:: The Ring of {C}onway Numbers in {M}izar
:: by Karol P\kak
::
:: Received December 12, 2023
:: Copyright (c) 2023-2026 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies MCART_1, FACIRC_1, RELAT_1, TARSKI, XBOOLE_0, LFUZZY_0, EUCLID_3,
PARTFUN1, SUBSET_1, XXREAL_0, ARYTM_3, FUNCT_1, ZFMISC_1, CARD_1,
ORDINAL2, ORDINAL1, PBOOLE, MEMBER_1, ARROW, ZFREFLE1, VECTSP_1,
ORDINAL4, ARYTM_1, FUNCOP_1, MATHMORP, GLIB_000, WAYBEL_3, COHSP_1,
ORDINAL6, SURREAL0, SURREALO, SURREALR;
notations TARSKI, XBOOLE_0, ZFMISC_1, XTUPLE_0, SUBSET_1, RELAT_1, FUNCT_1,
RELSET_1, PBOOLE, ORDINAL1, PARTFUN1, FUNCOP_1, ORDINAL2, CARD_1,
ORDINAL4, ORDINAL6, ORDINAL7, COHSP_1, SURREAL0, SURREALO;
constructors PBOOLE, RELSET_1, NEWTON, AFINSQ_1, ORDINAL6, ORDINAL7, COHSP_1,
SURREALO;
registrations XBOOLE_0, RELAT_1, SUBSET_1, XTUPLE_0, ORDINAL1, ORDINAL3,
FUNCT_1, NAT_1, ORDINAL6, FUNCOP_1, FOMODEL0, SURREAL0;
requirements BOOLE, SUBSET, NUMERALS;
definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, ORDINAL1, SURREAL0, SURREALO;
equalities ORDINAL1, SURREAL0, SURREALO;
expansions XTUPLE_0, SURREAL0, SURREALO;
theorems RELAT_1, TARSKI, ZFMISC_1, FUNCT_1, XBOOLE_0, XBOOLE_1, XTUPLE_0,
ORDINAL1, ORDINAL2, ORDINAL6, ORDINAL3, ORDINAL4, CARD_1, PARTFUN1,
FUNCOP_1, ORDINAL7, COHSP_1, SURREAL0, SURREALO;
schemes TARSKI, ORDINAL1, CLASSES1, XBOOLE_0, TARSKI_0;
begin :: Preliminaries
reserve A,B,C for Ordinal,
o for object,
x,y,z,t,r,l for Surreal,
X,Y for set;
definition
let f be Function;
redefine attr f is Function-yielding means :Def1: :: redefines Function-yielding (FUNCOP_1, definition)
rng f is functional;
compatibility
proof
thus f is Function-yielding implies rng f is functional
proof
assume A1:f is Function-yielding;
let o;
assume o in rng f;
then ex x be object st x in dom f & f.x=o by FUNCT_1:def 3;
hence thesis by A1;
end;
assume A2:rng f is functional;
now let o;
assume o in dom f;
then f.o in rng f by FUNCT_1:def 3;
hence f.o is Function by A2;
end;
hence thesis by FUNCOP_1:def 6;
end;
end;
registration
cluster c=-monotone Function-yielding for Sequence;
existence
proof
take F={};
for a, b being set st a in dom F & b in dom F & a c= b holds F.a c= F.b;
hence thesis by COHSP_1:def 11;
end;
end;
registration
let f be c=-monotone Function;
let X be set;
cluster f|X -> c=-monotone;
coherence
proof
for a, b being set st
a in dom (f|X) & b in dom (f|X) & a c= b holds
(f|X).a c= (f|X).b
proof
let a, b be set such that
A1:a in dom (f|X) & b in dom (f|X) and
A2: a c= b;
a in dom f & b in dom f by A1, RELAT_1:57;
then (f|X).a= f.a c= f.b by A1,A2,COHSP_1:def 11,FUNCT_1:47;
hence thesis by A1,FUNCT_1:47;
end;
hence thesis by COHSP_1:def 11;
end;
end;
registration
let f be c=-monotone Function-yielding Sequence;
cluster union rng f -> Function-like Relation-like;
coherence
proof
set U = union rng f;
U is Relation-like
proof
let x be object;assume x in U;
then consider Y be set such that
A1:x in Y & Y in rng f by TARSKI:def 4;
consider X be object such that
A2: X in dom f & f.X=Y by A1,FUNCT_1:def 3;
thus thesis by A1,A2,RELAT_1:def 1;
end;
then reconsider U as Relation;
for x,y1,y2 be object st [x,y1] in U & [x,y2] in U holds y1 = y2
proof
let x,y1,y2 be object such that A3: [x,y1] in U & [x,y2] in U;
consider Y1 be set such that
A4:[x,y1] in Y1 & Y1 in rng f by A3,TARSKI:def 4;
consider z1 be object such that
A5:z1 in dom f & f.z1 = Y1 by FUNCT_1:def 3,A4;
consider Y2 be set such that
A6:[x,y2] in Y2 & Y2 in rng f by A3,TARSKI:def 4;
consider z2 be object such that
A7:z2 in dom f & f.z2 = Y2 by FUNCT_1:def 3,A6;
reconsider z1,z2 as Ordinal by A5,A7;
z1 c= z2 or z2 c= z1;
then f.z1 c= f.z2 or f.z2 c= f.z1 by A5,A7,COHSP_1:def 11;
hence y1=y2 by FUNCT_1:def 1,A4,A6,A5,A7;
end;
hence thesis by FUNCT_1:def 1;
end;
end;
theorem Th1:
for f be c=-monotone Function-yielding Sequence
for o be object st o in dom union rng f
ex A st A in dom f & o in dom (f.A)
proof
let f be c=-monotone Function-yielding Sequence;
set U=union rng f;
let x be object;
assume x in dom U;
then [x,U.x] in U by FUNCT_1:def 2;
then consider Y be set such that
A1:[x,U.x] in Y & Y in rng f by TARSKI:def 4;
consider z be object such that
A2:z in dom f & f.z = Y by FUNCT_1:def 3,A1;
reconsider z as Ordinal by A2;
take z;
thus thesis by A2,A1,FUNCT_1:1;
end;
theorem Th2:
for f be c=-monotone Function-yielding Sequence
for A st A in dom f holds
dom (f.A) c= dom union rng f &
for o st o in dom (f.A) holds (f.A).o = (union rng f).o
proof
let f be c=-monotone Function-yielding Sequence;
set U=union rng f;
let A be Ordinal such that A1: A in dom f;
A2: f.A in rng f by A1,FUNCT_1:def 3;
thus dom (f.A) c= dom U
proof
let y be object;
assume y in dom (f.A);
then [y,f.A.y] in f.A by FUNCT_1:def 2;
then [y,f.A.y] in U by A2,TARSKI:def 4;
hence thesis by FUNCT_1:1;
end;
let y be object;
assume y in dom (f.A);
then [y,f.A.y] in f.A by FUNCT_1:def 2;
then [y,f.A.y] in U by A2,TARSKI:def 4;
hence thesis by FUNCT_1:1;
end;
theorem Th3:
for f be c=-monotone Function-yielding Sequence
for A be Ordinal,X be set st
for o st o in X ex B be Ordinal st o in dom (f.B) & B in A
holds
(union rng (f|A)).:X = (union rng f).:X
proof
let f be c=-monotone Function-yielding Sequence;
let A be Ordinal,X be set such that
A1:for x be object st x in X ex B be Ordinal st x in dom (f.B) & B in A;
union rng (f|A) c= union rng f by ZFMISC_1:77;
hence (union rng (f|A)).:X c= (union rng f).:X by RELAT_1:124;
let y be object such that A2:y in (union rng f).:X;
consider x be object such that
A3: x in dom (union rng f) & x in X & (union rng f).x=y
by A2,FUNCT_1:def 6;
consider B be Ordinal such that
A4: x in dom (f.B) & B in A by A3,A1;
A5: f.B<>{} by A4;
then B in dom f by FUNCT_1:def 2;
then A6: (f.B).x = (union rng f).x by A4,Th2;
A7:(f|A).B = f.B by A4,FUNCT_1:49;
then A8: B in dom (f|A) by A5,FUNCT_1:def 2;
then A9: dom ((f|A).B) c= dom union rng (f|A) by Th2;
y = (union rng (f|A)).x by A8,Th2,A7,A4,A3,A6;
hence thesis by A9,A7,A4,A3,FUNCT_1:def 6;
end;
begin :: Surreal Number Operators - Schemes
scheme MonoFvSExists{
O()->Ordinal,
D(Ordinal) -> set,
H(object,c=-monotone Function-yielding Sequence)->object}:
ex S be c=-monotone Function-yielding Sequence
st dom S = succ O() &
for A be Ordinal st A in succ O()
ex SA be ManySortedSet of D(A) st S.A = SA &
for o st o in D(A) holds SA.o = H(o,S|A)
provided
A1: for S be c=-monotone Function-yielding Sequence st
for A be Ordinal st A in dom S holds dom (S.A) = D(A)
for A be Ordinal,o st o in dom (S.A) holds H(o,S|A) = H(o,S)
and
A2: for A,B be Ordinal st A c= B holds D(A) c= D(B)
proof
defpred P[Ordinal] means
ex S be c=-monotone Function-yielding Sequence
st dom S = succ $1 &
for B be Ordinal st B in succ $1
ex fB be ManySortedSet of D(B) st S.B = fB &
for x be object st x in D(B) holds fB.x = H(x,S|B);
A3: for D be Ordinal st for C be Ordinal st C in D holds P[C] holds P[D]
proof
let D be Ordinal such that A4: for C be Ordinal st C in D holds P[C];
defpred R[object,object] means
$1 is Ordinal & $2 is c=-monotone Function-yielding Sequence
& for A be Ordinal st A=$1
for S be c=-monotone Function-yielding Sequence st $2=S holds
dom S = succ A &
for B be Ordinal st B in succ A
ex fB be ManySortedSet of D(B) st S.B = fB &
for x be object st x in D(B) holds fB.x = H(x,S|B);
A5:for e being object st e in D ex u being object st R[e,u]
proof
let e be object;
assume A6: e in D;
then reconsider E=e as Ordinal;
consider S be c=-monotone Function-yielding Sequence such that
A7: dom S = succ E &
for B be Ordinal st B in succ E
ex fB be ManySortedSet of D(B) st S.B = fB &
for x be object st x in D(B) holds fB.x = H(x,S|B) by A4,A6;
take S;
thus thesis by A7;
end;
consider f be Function such that
A8:dom f = D and
A9:for e being object st e in D holds R[e,f.e] from CLASSES1:sch 1(A5);
reconsider f as Sequence by A8,ORDINAL1:def 7;
A10: for A,B be Ordinal st A c= B & A in D & B in D holds f.A c= f.B
proof
let A,B be Ordinal such that
A11: A c= B & A in D & B in D;
A in succ B by A11,ORDINAL1:22;
then A12: succ A c= succ B by ORDINAL1:21;
reconsider fA=f.A,fB=f.B as c=-monotone Function-yielding Sequence
by A11,A9;
A13:dom fA = succ A by A9,A11;
A14:dom fB = succ B by A9,A11;
A15:for C be Ordinal st C in succ B
ex fC be ManySortedSet of D(C) st fB.C = fC &
for x be object st x in D(C) holds fC.x = H(x,fB|C) by A9,A11;
let x be object such that A16:x in f.A;
x in fA by A16;
then consider a,z be object such that
A17: x = [a,z] by RELAT_1:def 1;
A18: a in dom fA & fA.a = z by A17,FUNCT_1:1,A16;
then reconsider a as Ordinal;
defpred O[Ordinal] means $1 c= a implies fA.$1 = fB.$1;
A19: for F be Ordinal st for G be Ordinal st G in F holds O[G] holds O[F]
proof
let F be Ordinal such that A20: for G be Ordinal st G in F holds O[G];
assume A21: F c= a;
then A22: F in dom fA by A17,FUNCT_1:1,A16,ORDINAL1:12;
then consider fCF be ManySortedSet of D(F) such that
A23: fA.F = fCF and
A24: for x be object st x in D(F) holds fCF.x = H(x,fA|F)
by A13,A9,A11;
consider fBF be ManySortedSet of D(F) such that
A25: fB.F = fBF and
A26: for x be object st x in D(F) holds fBF.x = H(x,fB|F)
by A15,A12,A21,A13,A18,ORDINAL1:12;
A27: dom (fA|F) = F = dom (fB|F)
by RELAT_1:62,A22,A12,A13,A14,ORDINAL1:def 2;
for x be object st x in F holds (fA|F).x = (fB|F).x
proof
let x be object such that A28: x in F;
reconsider x as Ordinal by A28;
(fA|F).x = fA.x by A28,FUNCT_1:49
.=fB.x by A20,A28,A21,ORDINAL1:def 2;
hence thesis by A28,FUNCT_1:49;
end;
then A29: fA|F = fB|F by A27,FUNCT_1:2;
A30: dom fCF = D(F) = dom fBF by PARTFUN1:def 2;
for x be object st x in D(F) holds fCF.x = fBF.x
proof
let x be object such that A31: x in D(F);
fCF.x = H(x,fA|F) by A31,A24;
hence thesis by A31,A26,A29;
end;
hence thesis by A23,A25,A30,FUNCT_1:2;
end;
for D be Ordinal holds O[D] from ORDINAL1:sch 2(A19);
then fA.a = fB.a;
hence thesis by A12,A18,A13,A14,FUNCT_1:1,A17;
end;
A32: for A,B be set st A in D & B in D & A c= B holds
f.A c= f.B by A10;
for x be object st x in rng f holds x is Function
proof
let x be object;
assume x in rng f;
then ex y be object st y in dom f & f.y = x by FUNCT_1:def 3;
hence thesis by A8,A9;
end;
then reconsider f as c=-monotone Function-yielding Sequence
by A32,Def1,A8,COHSP_1:def 11,FUNCT_1:def 13;
A33:for x,y being set st x in rng f & y in rng f holds x,y are_c=-comparable
proof
let x,y be set such that A34:x in rng f & y in rng f;
consider A be object such that
A35: A in dom f & f.A =x by A34,FUNCT_1:def 3;
consider B be object such that
A36: B in dom f & f.B =y by A34,FUNCT_1:def 3;
reconsider A,B as Ordinal by A35,A36;
A c= B or B c= A;
then f.A c= f.B or f.B c= f.A by A10,A35,A36,A8;
hence thesis by A35,A36,XBOOLE_0:def 9;
end;
A37: for x be object st x in rng f holds x is Sequence
proof
let x be object;
assume x in rng f;
then ex y be object st y in dom f & f.y = x by FUNCT_1:def 3;
hence thesis by A8,A9;
end;
reconsider U = union rng f as Sequence by A33,A37,ORDINAL1:def 8,34;
A38: for x be object st x in rng U holds x is Function
proof
let x be object;
assume x in rng U;
then consider y be object such that
A39:y in dom U & U.y = x by FUNCT_1:def 3;
consider A be Ordinal such that
A40: A in dom f & y in dom (f.A) by Th1,A39;
A41: (f.A).y = U.y by A40,Th2;
reconsider fA=f.A as c=-monotone Function-yielding Sequence by A40,A8,A9;
dom fA = succ A by A8,A9,A40;
then reconsider y as Ordinal by A40;
for y be Ordinal st y in succ A
ex fC be ManySortedSet of D(y) st fA.y = fC &
for x be object st x in D(y) holds fC.x = H(x,fA|y) by A8,A9,A40;
hence thesis by A41,A39;
end;
then A42:rng U is functional by FUNCT_1:def 13;
A43:U is Function-yielding by A38,FUNCT_1:def 13;
A44: for a, b being set st a in dom U & b in dom U & a c= b holds U.a c= U.b
proof
let a,b be set such that A45:a in dom U & b in dom U & a c= b;
consider A be Ordinal such that
A46:A in dom f & a in dom (f.A) by Th1,A45;
consider B be Ordinal such that
A47:B in dom f & b in dom (f.B) by Th1,A45;
reconsider fA=f.A,fB=f.B as c=-monotone Function-yielding Sequence
by A46,A47,A8,A9;
A48: (f.B).b = U.b & (f.A).a = U.a by A46,A47,Th2;
per cases;
suppose B c= A;
then A49: fB c= fA by A10,A46,A47,A8;
[b,fB.b] in fB by A47,FUNCT_1:1;
then b in dom fA & fA.b = fB.b by A49,FUNCT_1:1;
hence thesis by A48,COHSP_1:def 11,A45,A46;
end;
suppose A c= B;
then A50: fA c= fB by A10,A46,A47,A8;
[a,fA.a] in fA by A46,FUNCT_1:1;
then a in dom fB & fB.a = fA.a by A50,FUNCT_1:1;
hence thesis by A45,A47,A48,COHSP_1:def 11;
end;
end;
then reconsider U as c=-monotone Function-yielding Sequence
by A43,COHSP_1:def 11;
A51: for A be Ordinal st A in dom U holds dom (U.A) = D(A)
proof
let y be Ordinal such that A52: y in dom U;
consider A be Ordinal such that
A53:A in dom f & y in dom (f.A) by Th1,A52;
A54:(f.A).y = U.y by A53,Th2;
reconsider fA=f.A as c=-monotone Function-yielding Sequence by A53,A8,A9;
A55: dom fA = succ A by A8,A9,A53;
reconsider y as Ordinal;
ex fC be ManySortedSet of D(y) st fA.y = fC &
for x be object st x in D(y) holds fC.x = H(x,fA|y) by A55,A8,A9,A53;
hence thesis by A54,PARTFUN1:def 2;
end;
A56: dom U c= D
proof
let a be Ordinal such that A57:a in dom U;
consider A be Ordinal such that
A58:A in dom f & a in dom (f.A) by Th1,A57;
reconsider fA=f.A as c=-monotone Function-yielding Sequence by A58,A8,A9;
A59:dom fA = succ A by A8,A9,A58;
succ A c= D by A58,A8,ORDINAL1:21;
hence a in D by A58,A59;
end;
A60: D c= dom U
proof
let A be Ordinal such that A61:A in D;
reconsider fA=f.A as c=-monotone Function-yielding Sequence by A61,A9;
dom fA = succ A &
for B be Ordinal st B in succ A
ex fB be ManySortedSet of D(B) st fA.B = fB &
for x be object st x in D(B) holds fB.x = H(x,fA|B) by A61,A9;
then succ A c= dom U by Th2,A61,A8;
hence thesis by ORDINAL1:21;
end;
then A62: D = dom U by A56,XBOOLE_0:def 10;
set UU = union rng U;
defpred S[object,object] means
($1 in dom UU implies $2=UU.$1) &
(not $1 in dom UU implies $2=H($1,U));
A63: for e being object st e in D(D) ex u being object st S[e,u]
proof
let e be object;
e in dom UU or not e in dom UU;
hence thesis;
end;
consider d be Function such that
A64:dom d = D(D) and
A65:for e being object st e in D(D) holds S[e,d.e] from CLASSES1:sch 1(A63);
reconsider d as ManySortedSet of D(D) by PARTFUN1:def 2,A64,RELAT_1:def 18;
set d1=1-->d, U1=U ^ d1;
A66: dom d1 = 1;
then A67: dom U1 = D +^ 1 = succ D by A62,ORDINAL4:def 1,ORDINAL2:31;
A68: rng U \/ {d} is functional by A42;
0 in 1 by TARSKI:def 1,CARD_1:49;
then A69: U1.(D +^ 0) = d1.0 = d by A66,A62,ORDINAL4:def 1,FUNCOP_1:7;
rng d1 = {d} by FUNCOP_1:8;
then A70: U1 is Function-yielding by A68,ORDINAL4:2;
for A,B be set st A in dom U1 & B in dom U1 & A c= B holds
U1.A c= U1.B
proof
let A,B be set such that A71: A in dom U1 & B in dom U1 & A c= B;
reconsider A,B as Ordinal by A71;
per cases by A67,A71,ORDINAL1:8;
suppose A72: A in D & B in D;
then U1.A = U.A & U1.B = U.B by ORDINAL4:def 1,A60;
hence thesis by A44,A60,A71,A72;
end;
suppose A73: A in D & B = D;
then A74:U1.A = U.A & U1.B = d by ORDINAL2:27,A69,ORDINAL4:def 1,A60;
consider E be Ordinal such that
A75:E in dom f & A in dom (f.E) by Th1,A73,A60;
reconsider fE=f.E as c=-monotone Function-yielding Sequence by A75,A8,A9;
dom fE = succ E by A8,A9,A75;
then consider fC be ManySortedSet of D(A) such that
A76: fE.A = fC and
for x be object st x in D(A) holds fC.x = H(x,fE|A) by A8,A9,A75;
A77: fC = U.A by A76,A75,Th2;
U.A c= d
proof
let x,y be object such that A78: [x,y] in U.A;
A79: dom fC = D(A) by PARTFUN1:def 2;
A80: x in dom fC & fC.x = y by A77,A78,FUNCT_1:1;
A81: D(A) c= D(D) by A2,A73,ORDINAL1:def 2;
dom fC c= dom UU by A77,Th2,A60,A73;
then d.x = UU.x by A80,A65,A79,A81;
hence thesis by A80,A77,Th2,A60,A73,FUNCT_1:1,A81,A79,A64;
end;
hence thesis by A74;
end;
suppose A = D & B in D;
hence thesis by A71, ORDINAL1:12;
end;
suppose A = D & B = D;
hence thesis;
end;
end;
then reconsider U1 as c=-monotone Function-yielding Sequence
by A70,COHSP_1:def 11;
take U1;
thus dom U1 = succ D by A67;
let B be Ordinal such that A82: B in succ D;
A83:dom U = dom U1 /\ dom U by A62,A67,XBOOLE_1:7,28;
for x being object st x in dom U holds U.x = U1.x
by ORDINAL4:def 1;
then A84: U = U1|D by A62,A83,FUNCT_1:46;
per cases by A82,ORDINAL1:8;
suppose A85:B in D;
consider E be Ordinal such that
A86:E in dom f & B in dom (f.E) by Th1,A85,A60;
reconsider fE=f.E as c=-monotone Function-yielding Sequence by A86,A8,A9;
dom fE = succ E by A8,A9,A86;
then consider fC be ManySortedSet of D(B) such that
A87: fE.B = fC and
A88: for x be object st x in D(B) holds fC.x = H(x,fE|B) by A8,A9,A86;
A89: fC = U.B by A87,A86,Th2;
take fC;
thus fC = U1.B by A85,ORDINAL4:def 1,A60,A89;
f.E in rng f by A86,FUNCT_1:def 3;
then f.E c= U by ZFMISC_1:74;
then A90: fE|B tolerates U|B by RELAT_1:76,PARTFUN1:54;
dom (fE|B)= B & dom (U|B)=B by A86,A85,ORDINAL1:def 2,A60,RELAT_1:62;
then A91: fE|B = U|B by A90,PARTFUN1:55;
let x be object such that A92:x in D(B);
U1|B = U1|D|B by A85,ORDINAL1:def 2,RELAT_1:74;
hence fC.x = H(x,U1|B) by A84,A88,A92,A91;
end;
suppose A93:B=D;
then reconsider dB=d as ManySortedSet of D(B);
take dB;
thus dB = U1.B by ORDINAL2:27,A69,A93;
let x be object;
assume A94: x in D(B);
per cases;
suppose A95: x in dom UU;
then A96: d.x = UU.x by A65,A93,A94;
consider A be Ordinal such that
A97: A in dom U & x in dom (U.A) by A95,Th1;
consider E be Ordinal such that
A98:E in dom f & A in dom (f.E) by A97,Th1;
reconsider fE=f.E as c=-monotone Function-yielding Sequence by A98,A8,A9;
dom fE = succ E by A8,A9,A98;
then consider fC be ManySortedSet of D(A) such that
A99: fE.A = fC and
A100: for x be object st x in D(A) holds fC.x = H(x,fE|A) by A8,A9,A98;
A101: dom fC =D(A) by PARTFUN1:def 2;
A102: fC = U.A by A99,A98,Th2;
then A103: fC.x = H(x,fE|A) by A97,A101,A100;
f.E in rng f by A98,FUNCT_1:def 3;
then A104: fE|A c= U|A by RELAT_1:76,ZFMISC_1:74;
dom (fE|A)= A & dom (U|A)=A by RELAT_1:62,A98,A97,ORDINAL1:def 2;
then fC.x = H(x,U|A) by A103,A104,PARTFUN1:54,55;
then fC.x = H(x,U1|D) by A84,A51,A1,A97;
hence dB.x = H(x,U1|B) by A96,A102,A97,Th2,A93;
end;
suppose not x in dom UU;
hence dB.x = H(x,U1|B) by A94,A65,A93,A84;
end;
end;
end;
for D be Ordinal holds P[D] from ORDINAL1:sch 2(A3);
hence thesis;
end;
scheme MonoFvSUniq{
O()->Ordinal,
D(Ordinal) -> set,
S1,S2() -> c=-monotone Function-yielding Sequence,
H(object,c=-monotone Function-yielding Sequence)->object}:
S1()|O() = S2()|O()
provided
A1: O() c= dom S1() & O() c= dom S2() and
A2: for A be Ordinal st A in O()
ex SA be ManySortedSet of D(A) st S1().A = SA &
for o st o in D(A) holds SA.o = H(o,S1()|A) and
A3: for A be Ordinal st A in O()
ex SA be ManySortedSet of D(A) st S2().A = SA &
for o st o in D(A) holds SA.o = H(o,S2()|A)
proof
defpred O[Ordinal] means $1 in O() implies S1().$1 = S2().$1;
A4: for F be Ordinal st for G be Ordinal st G in F holds O[G] holds O[F]
proof
let B be Ordinal such that A5: for G be Ordinal st G in B holds O[G];
assume A6: B in O();
then consider S1B be ManySortedSet of D(B) such that
A7: S1().B = S1B &
for x be object st x in D(B) holds S1B.x = H(x,S1()|B) by A2;
consider S2B be ManySortedSet of D(B) such that
A8:S2().B = S2B &
for x be object st x in D(B) holds S2B.x = H(x,S2()|B) by A3,A6;
A9:dom S1B = D(B) = dom S2B by PARTFUN1:def 2;
for x be object st x in D(B) holds S1B.x=S2B.x
proof
let x be object such that A10: x in D(B);
S1B<>{} & S2B<>{} by A10,A9;
then
B in dom S1() & B in dom S2() by A7,A8,FUNCT_1:def 2;
then A11: dom (S1()|B)=B =dom (S2()|B)
by RELAT_1:62,ORDINAL1:def 2;
for y be object st y in B holds (S1()|B).y=(S2()|B).y
proof
let y be object;
assume A12:y in B;
then reconsider Y=y as Ordinal;
A13:Y in O() by A12,A6,ORDINAL1:10;
thus (S1()|B).y = S1().Y by A12,FUNCT_1:49
.= S2().Y by A12,A5,A13
.=(S2()|B).y by A12,FUNCT_1:49;
end;
then A14: S1()|B = S2()|B by A11,FUNCT_1:2;
S1B.x = H(x,S1()|B) by A10,A7;
hence thesis by A14,A10,A8;
end;
hence thesis by A7,A8,A9,FUNCT_1:2;
end;
A15:for F be Ordinal holds O[F] from ORDINAL1:sch 2(A4);
A16: dom (S1()|O())=O() =dom (S2()|O()) by A1,RELAT_1:62;
for y be object st y in O() holds (S1()|O()).y=(S2()|O()).y
proof
let y be object;
assume A17:y in O();
then reconsider Y=y as Ordinal;
thus (S1()|O()).y = S1().Y by A17,FUNCT_1:49
.= S2().Y by A17,A15
.=(S2()|O()).y by A17,FUNCT_1:49;
end;
hence thesis by A16,FUNCT_1:2;
end;
begin :: The Opposite Surreal Number
definition
let A;
func No_opposite_op A -> ManySortedSet of Day A means :Def2:
ex S be c=-monotone Function-yielding Sequence st
dom S = succ A & it = S.A &
for B st B in succ A
ex SB be ManySortedSet of Day B st S.B = SB &
for o st o in Day B holds
SB.o = [union rng (S|B).:R_o, union rng(S|B).:L_o];
existence
proof
deffunc D(Ordinal) = Day $1;
deffunc H(object,c=-monotone Function-yielding Sequence) =
[(union rng $2).:R_$1, (union rng $2).:L_$1];
A1: for S be c=-monotone Function-yielding Sequence st
for A be Ordinal st A in dom S holds dom (S.A) = D(A) holds
for A be Ordinal,x be object st x in dom (S.A) holds H(x,S|A) = H(x,S)
proof
let S be c=-monotone Function-yielding Sequence such that
A2: for A be Ordinal st A in dom S holds dom (S.A) = D(A);
let A be Ordinal,x be object such that A3: x in dom (S.A);
S.A<>{} by A3;
then A4: A in dom S by FUNCT_1:def 2;
then A5:dom (S.A) = D(A) by A2;
then reconsider x as Surreal by A3;
A6:x = [L_x,R_x];
A7:for y be object st y in L_x\/R_x ex B be Ordinal st
y in dom (S.B) & B in A
proof
let y be object;
assume y in L_x\/R_x;
then consider O be Ordinal such that
A8: O in A & y in Day O by A5,A3,A6,SURREAL0:46;
dom (S.O) = Day O by A2,A4,A8,ORDINAL1:10;
hence thesis by A8;
end;
A9: now let y be object;
assume y in L_x;
then y in L_x \/ R_x by XBOOLE_0:def 3;
hence ex B be Ordinal st y in dom (S.B) & B in A by A7;
end;
now let y be object;
assume y in R_x;
then y in L_x \/ R_x by XBOOLE_0:def 3;
hence ex B be Ordinal st y in dom (S.B) & B in A by A7;
end;
then(union rng (S|A)).:R_x = (union rng S).:R_x by Th3;
hence thesis by A9,Th3;
end;
A10: for A,B be Ordinal st A c= B holds D(A) c= D(B) by SURREAL0:35;
consider S be c=-monotone Function-yielding Sequence such that
A11: dom S = succ A & for B be Ordinal st B in succ A
ex SB be ManySortedSet of D(B) st S.B = SB &
for x be object st x in D(B) holds SB.x = H(x,S|B)
from MonoFvSExists(A1,A10);
A in succ A by ORDINAL1:8;
then consider SB be ManySortedSet of D(A) such that
A12: S.A = SB &
for x be object st x in D(A) holds SB.x = H(x,S|A) by A11;
take SB,S;
thus thesis by A11,A12;
end;
uniqueness
proof
deffunc D(Ordinal) = Day $1;
deffunc H(object,c=-monotone Function-yielding Sequence) =
[(union rng $2).:R_$1, (union rng $2).:L_$1];
let O1,O2 be ManySortedSet of D(A);
given S1 be c=-monotone Function-yielding Sequence such that
A13:dom S1 = succ A & S1.A = O1 &
for B be Ordinal st B in succ A
ex SB be ManySortedSet of D(B) st S1.B = SB &
for x be object st x in D(B) holds
SB.x = H(x,S1|B);
given S2 be c=-monotone Function-yielding Sequence such that
A14:dom S2 = succ A & S2.A = O2 &
for B be Ordinal st B in succ A
ex SB be ManySortedSet of D(B) st S2.B = SB &
for x be object st x in D(B) holds
SB.x = H(x,S2|B);
A15: for B be Ordinal st B in succ A
ex SB be ManySortedSet of D(B) st S1.B = SB &
for x be object st x in D(B) holds SB.x = H(x,S1|B) by A13;
A16: for B be Ordinal st B in succ A
ex SB be ManySortedSet of D(B) st S2.B = SB &
for x be object st x in D(B) holds SB.x = H(x,S2|B) by A14;
A17: succ A c= dom S1 & succ A c= dom S2 by A13,A14;
S1|succ A = S2|succ A from MonoFvSUniq(A17,A15,A16);
then S1 = S2|succ A by A13;
hence thesis by A13,A14;
end;
end;
theorem Th4:
for S be c=-monotone Function-yielding Sequence st
for B st B in dom S
ex SB be ManySortedSet of Day B st S.B = SB &
for o st o in Day B holds
SB.o = [union rng (S|B).:R_o, union rng(S|B).:L_o]
holds
for A st A in dom S holds No_opposite_op A = S.A
proof
deffunc D(Ordinal) = Day $1;
deffunc H(object,c=-monotone Function-yielding Sequence) =
[(union rng $2).:R_$1, (union rng $2).:L_$1];
let S1 be c=-monotone Function-yielding Sequence such that
A1:for B be Ordinal st B in dom S1
ex SB be ManySortedSet of D(B) st S1.B = SB &
for x be object st x in D(B) holds SB.x = H(x,S1|B);
let A be Ordinal such that A2: A in dom S1;
A3:succ A c= dom S1 by A2,ORDINAL1:21;
consider S2 be c=-monotone Function-yielding Sequence such that
A4:dom S2 = succ A & S2.A = No_opposite_op A &
for B be Ordinal st B in succ A
ex SB be ManySortedSet of D(B) st S2.B = SB &
for x be object st x in D(B) holds
SB.x = H(x,S2|B) by Def2;
A5: for B be Ordinal st B in succ A
ex SB be ManySortedSet of D(B) st S1.B = SB &
for x be object st x in D(B) holds SB.x = H(x,S1|B) by A1,A3;
A6: for B be Ordinal st B in succ A
ex SB be ManySortedSet of D(B) st S2.B = SB &
for x be object st x in D(B) holds SB.x = H(x,S2|B) by A4;
A7: succ A c= dom S1 & succ A c= dom S2 by A2,ORDINAL1:21,A4;
A8: S1|succ A = S2|succ A from MonoFvSUniq(A7,A5,A6);
A in succ A by ORDINAL1:8;
hence thesis by A4,A8,FUNCT_1:49;
end;
theorem Th5:
for f be c=-monotone Function-yielding Sequence st
o in dom (f.B) & B in A
holds
o in dom union rng (f|A) &
(union rng (f|A)).o = (union rng f).o
proof
let f be c=-monotone Function-yielding Sequence;
assume A1:o in dom (f.B) & B in A;
A2: f.B<>{} by A1;
then B in dom f by FUNCT_1:def 2;
then A3: (f.B).o = (union rng f).o by A1,Th2;
A4:(f|A).B = f.B by A1,FUNCT_1:49;
then A5: B in dom (f|A) by A2,FUNCT_1:def 2;
then dom ((f|A).B) c= dom union rng (f|A) by Th2;
hence thesis by A3,A4,A5,Th2,A1;
end;
theorem Th6:
for f be c=-monotone Function-yielding Sequence
for A,B be Ordinal st o in dom (f.B) & B in A
holds
(union rng (f|A)).o = (union rng f).o
proof
let f be c=-monotone Function-yielding Sequence;
let A,B be Ordinal such that
A1: o in dom (f.B) & B in A;
A2: f.B<>{} by A1;
then B in dom f by FUNCT_1:def 2;
then A3: (f.B).o = (union rng f).o by A1,Th2;
A4:(f|A).B = f.B by A1,FUNCT_1:49;
(f|A).B = f.B by A1,FUNCT_1:49;
then B in dom (f|A) by A2,FUNCT_1:def 2;
hence thesis by A4,A3,Th2,A1;
end;
definition
let x;
func -x -> set equals
(No_opposite_op born x).x;
coherence;
end;
definition
let X be set;
func --X -> set means :Def4:
o in it iff ex x be Surreal st x in X & o = -x;
existence
proof
defpred P[object,object] means $1 is Surreal &
for x be Surreal st x=$1 holds $2=-x;
A1:for x,y,z being object st P[x,y] & P[x,z] holds y = z
proof
let x,y,z be object such that A2:P[x,y] & P[x,z];
reconsider xx=x as Surreal by A2;
y = -xx = z by A2;
hence thesis;
end;
consider M be set such that
A3:for x being object holds x in M iff
ex y being object st y in X & P[y,x] from TARSKI:sch 1(A1);
take M;
let y be object;
thus y in M implies ex x be Surreal st x in X & y = -x
proof
assume y in M;
then consider yy be object such that A4: yy in X & P[yy,y] by A3;
reconsider yy as Surreal by A4;
take yy;
thus thesis by A4;
end;
given x be Surreal such that A5:x in X & y = -x;
P[x,y] by A5;
hence thesis by A5,A3;
end;
uniqueness
proof
let M1,M2 be set such that
A6:for y be object holds
y in M1 iff ex x be Surreal st x in X & y = -x and
A7: for y be object holds
y in M2 iff ex x be Surreal st x in X & y = -x;
now let y be object;
y in M1 iff ex x be Surreal st x in X & y = -x by A6;
hence y in M1 iff y in M2 by A7;
end;
hence thesis by TARSKI:2;
end;
end;
theorem Th7:
-x = [-- R_x, --L_x]
proof
set A= born x;
consider S be c=-monotone Function-yielding Sequence such that
A1:dom S = succ A & No_opposite_op A = S.A and
A2:for B be Ordinal st B in succ A
ex SB be ManySortedSet of Day B st S.B = SB &
for x be object st x in Day B holds
SB.x = [union rng (S|B).:R_x, union rng(S|B).:L_x] by Def2;
consider SA be ManySortedSet of Day A such that
A3:S.A = SA and
A4:for x be object st x in Day A holds
SA.x = [union rng (S|A).:R_x, union rng(S|A).:L_x]
by A2,ORDINAL1:6;
set U = union rng (S|A);
x in Day A by SURREAL0:def 18;
then A5: [U.:R_x, U.:L_x] = -x by A1,A3,A4;
A6: -- R_x c= U.:R_x
proof
let y be object such that A7: y in -- R_x;
consider a be Surreal such that
A8:a in R_x & y = -a by Def4,A7;
set B = born a;
A9: a in L_x\/ R_x by A8,XBOOLE_0:def 3;
then A10: B in A by SURREALO:1;
A11:B in succ A by A9,SURREALO:1,ORDINAL1:8;
then consider SB be ManySortedSet of Day B such that
A12:S.B = SB and
for x be object st x in Day B holds
SB.x = [union rng (S|B).:R_x, union rng(S|B).:L_x] by A2;
A13:No_opposite_op B = S.B by A11,A1,A2,Th4;
A14: dom SB = Day B by PARTFUN1:def 2;
A15: a in Day B by SURREAL0:def 18;
A16:SB.a = (union rng S).a by Th2,A12,A14,A15,A11,A1;
(union rng S).a = U.a & a in dom U
by Th5,A10,A14,A15,A12;
hence thesis by FUNCT_1:def 6,A8,A16,A13,A12;
end;
U.:R_x c= -- R_x
proof
let y be object such that A17: y in U.:R_x;
consider a be object such that
A18: a in dom U & a in R_x & U.a = y by A17,FUNCT_1:def 6;
reconsider a as Surreal by A18,SURREAL0:def 16;
set B = born a;
A19: a in L_x\/ R_x by A18,XBOOLE_0:def 3;
then A20: B in A by SURREALO:1;
A21:B in succ A by SURREALO:1,A19,ORDINAL1:8;
then consider SB be ManySortedSet of Day B such that
A22:S.B = SB and
for x be object st x in Day B holds
SB.x = [union rng (S|B).:R_x, union rng(S|B).:L_x] by A2;
A23:No_opposite_op B = S.B by A21,A1,A2,Th4;
A24: dom SB = Day B by PARTFUN1:def 2;
A25: a in Day B by SURREAL0:def 18;
SB.a = (union rng S).a by Th2,A24,A25,A21,A22,A1;
then -a = y by A18,Th5,A20,A23,A24,A25,A22;
hence thesis by A18,Def4;
end;
then A26: U.:R_x = -- R_x by A6,XBOOLE_0:def 10;
A27: -- L_x c= U.:L_x
proof
let y be object such that A28: y in -- L_x;
consider a be Surreal such that
A29:a in L_x & y = -a by Def4,A28;
set B = born a;
A30: a in L_x\/ R_x by A29,XBOOLE_0:def 3;
then A31: B in A by SURREALO:1;
A32:B in succ A by A30,SURREALO:1,ORDINAL1:8;
then consider SB be ManySortedSet of Day B such that
A33:S.B = SB and
for x be object st x in Day B holds
SB.x = [union rng (S|B).:R_x, union rng(S|B).:L_x] by A2;
A34:No_opposite_op B = S.B by A32,A1,A2,Th4;
A35:dom SB = Day B by PARTFUN1:def 2;
A36:a in Day B by SURREAL0:def 18;
A37:SB.a = (union rng S).a by Th2,A33,A35,A36,A32,A1;
(union rng S).a = U.a & a in dom U by Th5,A31,A35,A36,A33;
hence thesis by FUNCT_1:def 6,A29,A37,A34,A33;
end;
U.:L_x c= -- L_x
proof
let y be object such that A38: y in U.:L_x;
consider a be object such that
A39: a in dom U & a in L_x & U.a = y by A38,FUNCT_1:def 6;
reconsider a as Surreal by A39,SURREAL0:def 16;
set B = born a;
A40: a in L_x\/ R_x by A39,XBOOLE_0:def 3;
then A41: B in A by SURREALO:1;
A42:B in succ A by SURREALO:1,A40,ORDINAL1:8;
then consider SB be ManySortedSet of Day B such that
A43:S.B = SB and
for x be object st x in Day B holds
SB.x = [union rng (S|B).:R_x, union rng(S|B).:L_x] by A2;
A44:No_opposite_op B = S.B by A42,A1,A2,Th4;
A45:dom SB = Day B by PARTFUN1:def 2;
A46:a in Day B by SURREAL0:def 18;
SB.a = (union rng S).a by Th2,A45,A46,A42,A43,A1;
then -a = y by A39,Th5,A41,A44,A45,A46,A43;
hence thesis by A39,Def4;
end;
hence thesis by A5,A26,A27,XBOOLE_0:def 10;
end;
Lm1: for D be Ordinal holds
(for x be Surreal st x in Day D holds -x is surreal & -x in Day D
& for x1 be Surreal st x1 = -x holds -x1 = x) &
for x,x1,y,y1 be Surreal st x in Day D & y in Day D
& x1=-x & y1=-y holds x <= y iff y1 <= x1
proof
defpred P[Ordinal] means
(for x be Surreal st x in Day $1 holds -x is surreal & -x in Day $1
& for x1 be Surreal st x1 = -x holds -x1 = x ) &
for x,x1,y,y1 be Surreal st x in Day $1 & y in Day $1
& x1=-x & y1=-y holds x <= y iff y1 <= x1;
A1: for D be Ordinal st for C be Ordinal st C in D holds P[C] holds P[D]
proof
let D be Ordinal such that
A2:for C be Ordinal st C in D holds P[C];
thus A3: for x be Surreal st x in Day D holds
-x is surreal & -x in Day D &
for x1 be Surreal st x1 = -x holds -x1 = x
proof
let x be Surreal such that A4: x in Day D;
A5:x=[L_x,R_x];
then A6: L_x << R_x by A4,SURREAL0:46;
A7: -- R_x << --L_x
proof
let l,r be Surreal such that
A8: l in -- R_x & r in -- L_x;
consider L be Surreal such that
A9: L in R_x & l=-L by A8,Def4;
consider R be Surreal such that
A10: R in L_x & r=-R by A8,Def4;
L in L_x \/ R_x by A9,XBOOLE_0:def 3;
then consider OL be Ordinal such that
A11: OL in D & L in Day OL by A4,A5,SURREAL0:46;
A12: not L <= R by A6,A10,A9;
R in L_x \/ R_x by A10,XBOOLE_0:def 3;
then consider OR be Ordinal such that
A13: OR in D & R in Day OR by A4,A5,SURREAL0:46;
set O=OL\/OR;
A14: Day OL c= Day O & Day OR c= Day O by SURREAL0:35,XBOOLE_1:7;
O in D by ORDINAL3:12,A11,A13;
hence thesis by A2,A12,A14,A9,A10,A13,A11;
end;
for y be object st y in -- R_x \/ -- L_x
ex O be Ordinal st O in D & y in Day O
proof
let y be object such that A15:y in -- R_x \/ -- L_x;
per cases by A15,XBOOLE_0:def 3;
suppose y in --R_x;
then consider L be Surreal such that
A16: L in R_x & y=-L by Def4;
L in L_x \/ R_x by A16,XBOOLE_0:def 3;
then consider OL be Ordinal such that
A17: OL in D & L in Day OL by A4,A5,SURREAL0:46;
P[OL] by A17,A2;
hence thesis by A16,A17;
end;
suppose y in --L_x;
then consider L be Surreal such that
A18: L in L_x & y=-L by Def4;
L in L_x \/ R_x by A18,XBOOLE_0:def 3;
then consider OL be Ordinal such that
A19: OL in D & L in Day OL by A4,A5,SURREAL0:46;
P[OL] by A19,A2;
hence thesis by A18,A19;
end;
end;
then [-- R_x, --L_x] in Day D by A7,SURREAL0:46;
hence -x is surreal & -x in Day D by Th7;
let x1 be Surreal such that A20:x1 = -x;
A21: x1 = [-- R_x, --L_x] by A20,Th7;
A22: -x1 = [-- R_x1, --L_x1] by Th7;
A23: -- R_x1 c= L_x
proof
let y be object;
assume y in --R_x1;
then consider y1 be Surreal such that
A24: y1 in R_x1 & - y1 = y by Def4;
consider y2 be Surreal such that
A25: y2 in L_x & - y2 = y1 by A21,A24,Def4;
y2 in L_x\/R_x by A25,XBOOLE_0:def 3;
then A26:born y2 in born x c= D by A4,SURREALO:1,SURREAL0:def 18;
y2 in Day born y2 by SURREAL0:def 18;
hence thesis by A25,A2,A26,A24;
end;
L_x c= -- R_x1
proof
let y be object;
assume A27: y in L_x;
then reconsider y as Surreal by SURREAL0:def 16;
y in L_x \/R_x by A27,XBOOLE_0:def 3;
then A28: born y in born x c= D & y in Day born y
by A4,SURREALO:1,SURREAL0:def 18;
then reconsider y1 = -y as Surreal by A2;
y1 in R_x1 by A21,A27,Def4;
then -y1 in -- R_x1 by Def4;
hence thesis by A28,A2;
end;
then A29: L_x = -- R_x1 by A23,XBOOLE_0:def 10;
A30: -- L_x1 c= R_x
proof
let y be object;assume y in --L_x1;
then consider y1 be Surreal such that
A31: y1 in L_x1 & - y1 = y by Def4;
consider y2 be Surreal such that
A32: y2 in R_x & - y2 = y1 by A21,A31,Def4;
y2 in L_x\/R_x by A32,XBOOLE_0:def 3;
then A33:born y2 in born x c= D by A4,SURREALO:1,SURREAL0:def 18;
y2 in Day born y2 by SURREAL0:def 18;
hence thesis by A32,A2,A33,A31;
end;
R_x c= -- L_x1
proof
let y be object;
assume A34: y in R_x;
then reconsider y as Surreal by SURREAL0:def 16;
y in L_x \/R_x by A34,XBOOLE_0:def 3;
then A35: born y in born x c= D & y in Day born y
by A4,SURREALO:1,SURREAL0:def 18;
then reconsider y1 = -y as Surreal by A2;
y1 in L_x1 by A21,A34,Def4;
then -y1 in -- L_x1 by Def4;
hence thesis by A35,A2;
end;
then R_x =--L_x1 by A30,XBOOLE_0:def 10;
hence thesis by A22,A29;
end;
defpred O[Ordinal] means
for x,x1,y,y1 be Surreal st x in Day D & y in Day D
& x1=-x & y1=-y & born x (+) born y c= $1 holds x <= y iff y1 <= x1;
A36: for E be Ordinal st for F be Ordinal st F in E holds O[F] holds O[E]
proof
let E be Ordinal such that
A37:for C be Ordinal st C in E holds O[C];
A38: for x,x1,y,y1 be Surreal st x in Day D & y in Day D
& x1=-x & y1=-y & born x (+) born y c= E & x <= y holds y1 <= x1
proof
let x,x1,y,y1 be Surreal such that A39:x in Day D & y in Day D
& x1=-x & y1=-y & born x (+) born y c= E & x <= y;
A40: -x = [-- R_x, --L_x] & -y = [-- R_y, --L_y] by Th7;
A41: x in Day born x & born x c= D by A39,SURREAL0:def 18;
A42: y in Day born y & born y c= D by A39,SURREAL0:def 18;
A43: L_x << {y} & {x} << R_y by A39,SURREAL0:43;
A44: {y1} << R_x1
proof
let r,l1 be Surreal such that A45: r in {y1} & l1 in R_x1;
consider l be Surreal such that
A46: l in L_x & l1 = -l by A45,A39,A40,Def4;
A47: l in L_x \/ R_x by XBOOLE_0:def 3,A46;
then A48: born l in born x by SURREALO:1;
A49: born l (+) born y in born x (+)born y
by A47,SURREALO:1,ORDINAL7:94;
A50:l in Day born l c= Day D
by A41,A48,SURREAL0:def 18,35,ORDINAL1:def 2;
A51:r=y1 by A45,TARSKI:def 1;
y in {y} by TARSKI:def 1;
then not y <= l by A43,A46;
hence thesis by A50,A49,A39,A37,A46,A51;
end;
L_y1 << {x1}
proof
let l1,r be Surreal such that A52:l1 in L_y1 & r in {x1};
consider l be Surreal such that
A53: l in R_y & l1 = -l by A52,A39,A40,Def4;
A54: l in L_y \/ R_y by XBOOLE_0:def 3,A53;
then A55: born l in born y by SURREALO:1;
A56:born l (+) born x in born x (+)born y
by A54,SURREALO:1,ORDINAL7:94;
A57: l in Day born l c= Day D
by A42,A55,SURREAL0:def 18,35,ORDINAL1:def 2;
A58:r=x1 by A52,TARSKI:def 1;
x in {x} by TARSKI:def 1;
then not l <= x by A43,A53;
hence thesis by A58,A57,A56,A39,A37,A53;
end;
hence thesis by A44,SURREAL0:43;
end;
let x,x1,y,y1 be Surreal such that A59:x in Day D & y in Day D
& x1=-x & y1=-y & born x (+) born y c= E;
thus x <= y implies y1 <= x1 by A59,A38;
A60:x in Day born x by SURREAL0:def 18;
born x c= D by A59,SURREAL0:def 18;
then born x = D or born x in D by ORDINAL1:11,XBOOLE_0:def 8;
then x1 in Day born x by A60,A3,A2,A59;
then A61:born x1 c= born x by SURREAL0:def 18;
A62:x1 in Day born x1 by SURREAL0:def 18;
A63:x1 in Day D by A3,A59;
then born x1 c= D by SURREAL0:def 18;
then A64:born x1 = D or born x1 in D by ORDINAL1:11,XBOOLE_0:def 8;
A65: -x1=x by A3,A59;
then x in Day born x1 by A62,A64,A3,A2;
then born x c= born x1 by SURREAL0:def 18;
then A66: born x = born x1 by A61,XBOOLE_0:def 10;
A67:y in Day born y by SURREAL0:def 18;
born y c= D by A59,SURREAL0:def 18;
then born y = D or born y in D by ORDINAL1:11,XBOOLE_0:def 8;
then y1 in Day born y by A67,A3,A2,A59;
then A68:born y1 c= born y by SURREAL0:def 18;
A69:y1 in Day born y1 by SURREAL0:def 18;
A70: y1 in Day D by A3,A59;
then born y1 c= D by SURREAL0:def 18;
then A71: born y1 = D or born y1 in D by ORDINAL1:11,XBOOLE_0:def 8;
A72: -y1=y by A3,A59;
then y in Day born y1 by A69,A71,A3,A2;
then born y c= born y1 by SURREAL0:def 18;
then born y = born y1 by A68,XBOOLE_0:def 10;
hence thesis by A59,A38,A66,A65,A72,A70,A63;
end;
A73:for E be Ordinal holds O[E] from ORDINAL1:sch 2(A36);
let x,x1,y,y1 be Surreal such that A74:x in Day D & y in Day D
& x1=-x & y1=-y;
O[born x (+) born y] by A73;
hence thesis by A74;
end;
for E be Ordinal holds P[E] from ORDINAL1:sch 2(A1);
hence thesis;
end;
registration
let x;
cluster -x -> surreal;
coherence
proof
x in Day born x by SURREAL0:def 18;
hence thesis by Lm1;
end;
end;
registration
let X be set;
cluster --X -> surreal-membered;
coherence
proof
let y be object;
assume y in --X;
then ex x be Surreal st x in X & y = -x by Def4;
hence thesis;
end;
end;
theorem Th8:
L_(-x) = --R_x & R_(-x) = --L_x
proof
-x = [-- R_x, --L_x] by Th7;
hence thesis;
end;
::$N Conway Ch.1 Th.4(ii)
theorem Th9:
- -x = x
proof
x in Day born x by SURREAL0:def 18;
hence thesis by Lm1;
end;
registration
let x;
reduce - - x to x;
reducibility by Th9;
end;
theorem Th10:
x <= y iff -y <= -x
proof
set Bx=born x,By=born y,B=Bx\/By;
A1:Day Bx c= Day B & Day By c= Day B by SURREAL0:35,XBOOLE_1:7;
x in Day Bx & y in Day By by SURREAL0:def 18;
hence thesis by A1,Lm1;
end;
theorem
for x be Surreal, D be Ordinal st x in Day D holds -x in Day D by Lm1;
theorem Th12:
born x = born -x
proof
x in Day born x & -x in Day born -x by SURREAL0:def 18;
then -x in Day born x & - - x in Day born -x & - -x = x by Lm1;
then born -x c= born x & born x c= born -x by SURREAL0:def 18;
hence thesis by XBOOLE_0:def 10;
end;
theorem Th13:
born_eq x = born_eq -x
proof
consider y such that
A1: born y = born_eq x & y ==x by SURREALO:def 5;
A2: -y == - x by A1,Th10;
A3: born y = born -y by Th12;
for z st z == -x holds born y c= born z
proof
let z;
assume z == -x;
then -z == - -x = x by Th10;
then born_eq x c= born -z by SURREALO:def 5;
hence thesis by A1,Th12;
end;
hence thesis by A1,A2,A3,SURREALO:def 5;
end;
theorem
x in born_eq_set y implies -x in born_eq_set -y
proof
assume A1:x in born_eq_set y;
then x == y by SURREALO:def 6;
then A2: -x == -y by Th10;
x in Day born_eq y by A1,SURREALO:def 6;
then born -x = born x c= born_eq y =born_eq -y
by SURREAL0:def 18,Th13,Th12;
then -x in Day (born -x) c= Day (born_eq -y) by SURREAL0:def 18,35;
hence thesis by A2,SURREALO:def 6;
end;
theorem Th15:
for X be surreal-membered set
holds -- -- X = X
proof
let X be surreal-membered set;
thus -- -- X c= X
proof
let x be object;
assume x in -- -- X;
then consider y be Surreal such that A1: y in --X & x = -y by Def4;
ex z be Surreal st z in X & y = -z by A1,Def4;
hence thesis by A1;
end;
let x be object;
assume A2: x in X;
then reconsider x as Surreal by SURREAL0:def 16;
-x in --X by A2,Def4;
then - -x in -- --X by Def4;
hence thesis;
end;
theorem Th16:
card (--X) c= card X
proof
defpred P[object,object] means for x st x=$1 holds $2 = -x;
A1:o in --X implies ex u being object st P[o,u]
proof
assume o in --X;
then consider x such that
A2: x in X & o = -x by Def4;
take x;
let y;
thus thesis by A2;
end;
consider f be Function such that
A3:dom f = --X and
A4: for o being object st o in --X holds P[o,f.o] from CLASSES1:sch 1(A1);
A5:rng f c= X
proof
let y be object;
assume y in rng f;
then consider x be object such that
A6: x in dom f & f.x=y by FUNCT_1:def 3;
consider z be Surreal such that
A7: z in X & x = -z by A3,A6,Def4;
f.x = - -z =z by A6,A7,A3,A4;
hence thesis by A7,A6;
end;
f is one-to-one
proof
let x1,x2 be object such that A8: x1 in dom f & x2 in dom f & f.x1 = f.x2;
consider y1 be Surreal such that
A9:y1 in X & x1 = -y1 by A8,A3,Def4;
consider y2 be Surreal such that
A10: y2 in X & x2 = -y2 by A8,A3,Def4;
f.x1 = - -y1 =y1 & f.x2 = - -y2 =y2 by A10,A9,A3,A8,A4;
hence thesis by A8,A9,A10;
end;
hence thesis by CARD_1:10,A3,A5;
end;
theorem
for X be surreal-membered set holds
card X = card --X
proof
let X be surreal-membered set;
-- -- X = X by Th15;
then card (--X) c= card X c= card (--X) by Th16;
hence thesis by XBOOLE_0:def 10;
end;
Lm2:
for X,Y be surreal-membered set st X <<= Y holds --Y <<= --X
proof
let X,Y be surreal-membered set such that A1: X <<= Y;
let l1,r1 be Surreal such that A2: l1 in --X & r1 in --Y;
consider l be Surreal such that A3: l in X & l1 = -l by A2,Def4;
ex r be Surreal st r in Y & r1 = -r by A2,Def4;
hence thesis by A3,A1,Th10;
end;
theorem
for X,Y be surreal-membered set
holds X <<= Y iff --Y <<= --X
proof
let X,Y be surreal-membered set;
-- -- X = X & -- -- Y = Y by Th15;
hence thesis by Lm2;
end;
Lm3:
for X,Y be surreal-membered set st X << Y holds --Y << --X
proof
let X,Y be surreal-membered set such that A1: X << Y;
let l1,r1 be Surreal such that A2: l1 in --Y & r1 in --X;
A3:ex l be Surreal st l in Y & l1 = -l by A2,Def4;
ex r be Surreal st r in X & r1 = -r by A2,Def4;
hence thesis by A3,A1,Th10;
end;
theorem
for X,Y be surreal-membered set
holds X << Y iff --Y << --X
proof
let X,Y be surreal-membered set;
-- -- X = X & -- -- Y = Y by Th15;
hence thesis by Lm3;
end;
theorem Th20:
for X1,X2 be set holds
--(X1\/X2) = (-- X1) \/ (--X2)
proof
let X1,X2 be set;
thus --(X1\/X2) c= (-- X1) \/ (--X2)
proof
let xy be object;
assume xy in --(X1\/X2);
then consider x be Surreal such that
A1: x in X1\/X2 & xy= -x by Def4;
x in X1 or x in X2 by A1,XBOOLE_0:def 3;
then xy in (--X1) or xy in (--X2) by A1,Def4;
hence thesis by XBOOLE_0:def 3;
end;
let xy be object;
assume xy in (-- X1) \/ (--X2);
then per cases by XBOOLE_0:def 3;
suppose xy in --X1;
then consider x be Surreal such that
A2: x in X1 & xy = -x by Def4;
x in X1\/X2 by A2,XBOOLE_0:def 3;
hence thesis by A2,Def4;
end;
suppose xy in --X2;
then consider x be Surreal such that
A3: x in X2 & xy=-x by Def4;
x in X1\/X2 by A3,XBOOLE_0:def 3;
hence thesis by A3,Def4;
end;
end;
theorem Th21:
{-x} = --{x}
proof
o in {-x} implies o in --{x}
proof
assume o in {-x};
then o =-x & x in {x} by TARSKI:def 1;
hence thesis by Def4;
end;
hence {-x} c= --{x} by TARSKI:def 3;
let xy be object;
assume xy in --{x};
then consider y such that
A1: y in {x} & xy = -y by Def4;
x=y by A1,TARSKI:def 1;
hence thesis by A1,TARSKI:def 1;
end;
theorem Th22:
-- {} = {}
proof
assume -- {} <> {};
then consider x be object such that
A1: x in --{} by XBOOLE_0:def 1;
ex y st y in {} & x=-y by A1,Def4;
hence thesis;
end;
theorem Th23:
- 0_No = 0_No
proof
- 0_No = [--R_0_No, --L_0_No] by Th7;
hence thesis by Th22;
end;
registration
reduce - 0_No to 0_No;
reducibility by Th23;
end;
theorem
x==0_No iff -x == 0_No by Th23,Th10;
definition
let A be Ordinal;
func Triangle A -> Subset of [:Day A,Day A:] means :Def5:
for x,y be Surreal holds
[x,y] in it iff born x (+) born y c= A;
existence
proof
set IT = {[x,y] where x,y is Element of Day A: born x (+) born y c= A};
IT c= [:Day A,Day A:]
proof
let a be object;
assume a in IT;
then ex x1,y1 be Element of Day A st
a=[x1,y1] & born x1 (+) born y1 c= A;
hence thesis by ZFMISC_1:87;
end;
then reconsider IT as Subset of [:Day A,Day A:];
take IT;
let x,y be Surreal;
thus [x,y] in IT implies born x (+) born y c= A
proof
assume [x,y] in IT;
then consider x1,y1 be Element of Day A such that
A1: [x,y]=[x1,y1] & born x1 (+) born y1 c= A;
x=x1& y = y1 by A1,XTUPLE_0:1;
hence thesis by A1;
end;
A2: born x c= born x (+) born y & born y c= born x (+) born y
by ORDINAL7:86;
assume A3: born x (+) born y c= A;
then born x c= A & born y c= A by A2,XBOOLE_1:1;
then x in Day born x c= Day A & y in Day born y c= Day A
by SURREAL0:35,SURREAL0:def 18;
hence thesis by A3;
end;
uniqueness
proof
let IT1,IT2 be Subset of [:Day A,Day A:] such that
A4:for x,y be Surreal holds
[x,y] in IT1 iff born x (+) born y c= A and
A5:for x,y be Surreal holds
[x,y] in IT2 iff born x (+) born y c= A;
let x,y be object;
thus [x,y] in IT1 implies [x,y] in IT2
proof
assume A6:[x,y] in IT1;
then reconsider x1=x,y1=y as Element of Day A by ZFMISC_1:87;
born x1 (+) born y1 c= A by A6,A4;
hence thesis by A5;
end;
assume A7:[x,y] in IT2;
then reconsider x1=x,y1=y as Element of Day A by ZFMISC_1:87;
born x1 (+) born y1 c= A by A7,A5;
hence thesis by A4;
end;
end;
registration
let A be Ordinal;
cluster Triangle A -> non empty;
coherence
proof
set x = the Element of Day A,y = the Element of Day 0;
born y c= 0 by SURREAL0:def 18;
then A1:born y = 0;
born x c= A by SURREAL0:def 18;
then born x (+) born y c= A by A1,ORDINAL7:69;
hence thesis by Def5;
end;
end;
theorem Th25:
for A,B be Ordinal st A c= B holds Triangle A c= Triangle B
proof
let A,B be Ordinal such that A1: A c= B;
let x,y be object such that A2: [x,y] in Triangle A;
reconsider x,y as Element of Day A by A2,ZFMISC_1:87;
born x (+) born y c= A by A2,Def5;
then born x (+) born y c= B by A1,XBOOLE_1:1;
hence thesis by Def5;
end;
begin :: The Sum of Surreal Numbers
definition
let A be Ordinal;
func No_sum_op A -> ManySortedSet of Triangle A means :Def6:
ex S be c=-monotone Function-yielding Sequence st
dom S = succ A & it = S.A &
for B be Ordinal st B in succ A
ex SB be ManySortedSet of Triangle B st S.B = SB &
for x be object st x in Triangle B holds
SB.x = [union rng (S|B).:([:L_L_x,{R_x}:]\/[:{L_x},L_R_x:] ),
union rng(S|B).:([:R_L_x,{R_x}:]\/[:{L_x},R_R_x:] )];
existence
proof
deffunc D(Ordinal) = Triangle $1;
deffunc H(object,c=-monotone Function-yielding Sequence) =
[union rng $2.:([:L_L_$1,{R_$1}:]\/[:{L_$1},L_R_$1:] ),
union rng $2.:([:R_L_$1,{R_$1}:]\/[:{L_$1},R_R_$1:])];
A1: for A,B be Ordinal st A c= B holds D(A) c= D(B) by Th25;
A2: for S be c=-monotone Function-yielding Sequence st
for A be Ordinal st A in dom S holds dom (S.A) = D(A)
for A be Ordinal,x be object st x in dom (S.A) holds H(x,S|A) = H(x,S)
proof
let S be c=-monotone Function-yielding Sequence such that
A3:for A be Ordinal st A in dom S holds dom (S.A) = D(A);
let A be Ordinal,a be object such that A4: a in dom (S.A);
S.A <>{} by A4;
then A5:A in dom S by FUNCT_1:def 2;
then A6: dom (S.A) = D(A) by A3;
then consider x,y be object such that
A7:x in Day A & y in Day A & a=[x,y] by A4,ZFMISC_1:def 2;
reconsider x,y as Surreal by A7;
A8: born x (+) born y c= A by A7,A6,A4,Def5;
set LL=[:L_L_a,{R_a}:]\/[:{L_a},L_R_a:];
for b be object st b in LL
ex B be Ordinal st b in dom (S.B) & B in A
proof
let b be object such that A9:b in LL;
per cases by A9,XBOOLE_0:def 3;
suppose b in [:L_L_a,{R_a}:];
then consider l,r be object such that
A10: l in L_x & r in {y} & b = [l,r] by A7,ZFMISC_1:def 2;
reconsider l as Surreal by A10,SURREAL0:def 16;
set B = born l (+) born y;
take B;
l in L_x \/ R_x by A10,XBOOLE_0:def 3;
then A11: B in born x (+) born y by ORDINAL7:94,SURREALO:1;
then A12:dom (S.B) = D(B) by A8,A3,A5,ORDINAL1:10;
[l,y] in D(B) by Def5;
hence thesis by A12,A10,A11,A8,TARSKI:def 1;
end;
suppose b in [:{L_a},L_R_a:];
then consider l,r be object such that
A13: l in {x} & r in L_y & b = [l,r] by A7,ZFMISC_1:def 2;
reconsider r as Surreal by A13,SURREAL0:def 16;
set B = born x (+) born r;
take B;
r in L_y \/ R_y by A13,XBOOLE_0:def 3;
then A14: B in born x (+) born y by ORDINAL7:94,SURREALO:1;
then A15:dom (S.B) = D(B) by A8,A3,A5,ORDINAL1:10;
[x,r] in D(B) by Def5;
hence thesis by A15,A13,A14,A8,TARSKI:def 1;
end;
end;
then A16: (union rng (S|A)).:LL = (union rng S).:LL by Th3;
set RR=[:R_L_a,{R_a}:]\/[:{L_a},R_R_a:];
for b be object st b in RR
ex B be Ordinal st b in dom (S.B) & B in A
proof
let b be object such that A17:b in RR;
per cases by A17,XBOOLE_0:def 3;
suppose b in [:R_L_a,{R_a}:];
then consider l,r be object such that
A18: l in R_x & r in {y} & b = [l,r] by A7,ZFMISC_1:def 2;
reconsider l as Surreal by A18,SURREAL0:def 16;
set B = born l (+) born y;
take B;
l in L_x \/ R_x by A18,XBOOLE_0:def 3;
then A19: B in born x (+) born y by ORDINAL7:94,SURREALO:1;
then A20:dom (S.B) = D(B) by A8,A3,A5,ORDINAL1:10;
[l,y] in D(B) by Def5;
hence thesis by A20,A18,A19,A8,TARSKI:def 1;
end;
suppose b in [:{L_a},R_R_a:];
then consider l,r be object such that
A21: l in {x} & r in R_y & b = [l,r] by A7,ZFMISC_1:def 2;
reconsider r as Surreal by A21,SURREAL0:def 16;
set B = born x (+) born r;
take B;
r in L_y \/ R_y by A21,XBOOLE_0:def 3;
then A22: B in born x (+) born y by ORDINAL7:94,SURREALO:1;
then A23:dom (S.B) = D(B) by A8,A3,A5,ORDINAL1:10;
[x,r] in D(B) by Def5;
hence thesis by A23,A21,A22,A8,TARSKI:def 1;
end;
end;
hence thesis by A16,Th3;
end;
consider S be c=-monotone Function-yielding Sequence such that
A24: dom S = succ A and
A25: for A1 be Ordinal st A1 in succ A
ex SA1 be ManySortedSet of D(A1) st S.A1 = SA1 &
for x be object st x in D(A1) holds SA1.x = H(x,S|A1)
from MonoFvSExists(A2,A1);
consider SA be ManySortedSet of D(A) such that
A26:S.A = SA and
for x be object st x in D(A) holds SA.x = H(x,S|A) by A25,ORDINAL1:6;
take SA,S;
thus thesis by A24,A26,A25;
end;
uniqueness
proof
deffunc D(Ordinal) = Triangle $1;
deffunc H(object,c=-monotone Function-yielding Sequence) =
[union rng $2.:([:L_L_$1,{R_$1}:]\/[:{L_$1},L_R_$1:] ),
union rng $2.:([:R_L_$1,{R_$1}:]\/[:{L_$1},R_R_$1:])];
let it1,it2 be ManySortedSet of Triangle A;
given S1 be c=-monotone Function-yielding Sequence such that
A27:dom S1 = succ A & it1 = S1.A and
A28:for B be Ordinal st B in succ A
ex SB be ManySortedSet of D(B) st S1.B = SB &
for x be object st x in D(B) holds
SB.x = H(x,S1|B);
given S2 be c=-monotone Function-yielding Sequence such that
A29:dom S2 = succ A & it2 = S2.A and
A30:for B be Ordinal st B in succ A
ex SB be ManySortedSet of D(B) st S2.B = SB &
for x be object st x in D(B) holds SB.x = H(x,S2|B);
A31:succ A c= dom S1 & succ A c= dom S2 by A27,A29;
S1|succ A = S2|succ A from MonoFvSUniq(A31,A28,A30);
then S1|succ A = S2 by A29;
hence thesis by A27,A29;
end;
end;
theorem Th26:
for S be c=-monotone Function-yielding Sequence st
for B be Ordinal st B in dom S
ex SB be ManySortedSet of Triangle B st S.B = SB &
for x be object st x in Triangle B holds
SB.x = [union rng(S|B).:([:L_L_x,{R_x}:]\/[:{L_x},L_R_x:]),
union rng(S|B).:([:R_L_x,{R_x}:]\/[:{L_x},R_R_x:])]
holds
for A be Ordinal st A in dom S holds No_sum_op A = S.A
proof
deffunc D(Ordinal) = Triangle $1;
deffunc H(object,c=-monotone Function-yielding Sequence) =
[union rng $2.:([:L_L_$1,{R_$1}:]\/[:{L_$1},L_R_$1:] ),
union rng $2.:([:R_L_$1,{R_$1}:]\/[:{L_$1},R_R_$1:])];
let S be c=-monotone Function-yielding Sequence such that
A1: for B be Ordinal st B in dom S
ex SB be ManySortedSet of D(B) st S.B = SB &
for x be object st x in D(B) holds
SB.x = H(x,S|B);
let A be Ordinal such that A2: A in dom S;
consider S1 be c=-monotone Function-yielding Sequence such that
A3:dom S1 = succ A & No_sum_op A = S1.A and
A4:for B be Ordinal st B in succ A
ex SB be ManySortedSet of D(B) st S1.B = SB &
for x be object st x in D(B) holds SB.x = H(x,S1|B) by Def6;
A5:succ A c= dom S by A2,ORDINAL1:21;
A6: for B be Ordinal st B in succ A
ex SB be ManySortedSet of D(B) st S.B = SB &
for x be object st x in D(B) holds SB.x = H(x,S|B) by A1,A5;
A7: succ A c= dom S & succ A c= dom S1 by A2,ORDINAL1:21,A3;
A8: S|succ A = S1|succ A from MonoFvSUniq(A7,A6,A4);
A in succ A by ORDINAL1:8;
hence thesis by A3,A8,FUNCT_1:49;
end;
definition
let x,y be Surreal;
func x + y -> set equals
(No_sum_op (born x (+) born y)).[x,y];
coherence;
end;
definition
let X,Y be set;
func X ++ Y -> set means :Def8:
o in it iff ex x,y be Surreal st x in X & y in Y & o = x + y;
existence
proof
defpred P[object,object] means L_$1 is Surreal & $1`2 is Surreal &
for x,y be Surreal st x=L_$1 & y = $1`2 holds $2=x+y;
A1:for x,y,z being object st P[x,y] & P[x,z] holds y = z
proof
let x,y,z be object such that A2:P[x,y] & P[x,z];
reconsider x1=L_x,x2=R_x as Surreal by A2;
y = x1 + x2 = z by A2;
hence thesis;
end;
consider M be set such that
A3:for x being object holds x in M iff
ex y being object st y in [:X,Y:] & P[y,x] from TARSKI:sch 1(A1);
take M;
let a be object;
thus a in M implies ex x,y be Surreal st x in X & y in Y & a = x + y
proof
assume a in M;
then consider yy be object such that A4: yy in [:X,Y:] & P[yy,a] by A3;
reconsider y1=L_yy,y2=R_yy as Surreal by A4;
take y1,y2;
yy is pair by A4;
hence thesis by A4,ZFMISC_1:87;
end;
given x,y be Surreal such that A5:x in X & y in Y & a = x + y;
A6:[x,y] in [:X,Y:] by A5,ZFMISC_1:87;
P[[x,y],a] by A5;
hence thesis by A6,A3;
end;
uniqueness
proof
let M1,M2 be set such that
A7:for z be object holds
z in M1 iff ex x,y be Surreal st x in X & y in Y & z = x + y and
A8: for z be object holds
z in M2 iff ex x,y be Surreal st x in X & y in Y & z = x + y;
now let z be object;
z in M1 iff ex x,y be Surreal st x in X & y in Y & z = x + y by A7;
hence z in M1 iff z in M2 by A8;
end;
hence thesis by TARSKI:2;
end;
end;
theorem Th27:
for X be set holds X++{} = {}
proof
let X be set;
assume X++{} <> {};
then consider x be object such that
A1:x in X++{} by XBOOLE_0:def 1;
ex a,b be Surreal st a in X & b in {}& x=a+b by Def8,A1;
hence thesis;
end;
theorem Th28:
for x,y be Surreal holds
x + y = [(L_x ++ {y})\/({x} ++ L_y), (R_x ++ {y}) \/({x} ++ R_y)]
proof
let x,y be Surreal;
set Bx=born x,By=born y,A=Bx (+)By;
consider S be c=-monotone Function-yielding Sequence such that
A1: dom S = succ A & No_sum_op A = S.A and
A2: for B be Ordinal st B in succ A
ex SB be ManySortedSet of Triangle B st S.B = SB &
for x be object st x in Triangle B holds
SB.x = [union rng (S|B).:([:L_L_x,{R_x}:]\/[:{L_x},L_R_x:] ),
union rng(S|B).:([:R_L_x,{R_x}:]\/[:{L_x},R_R_x:] )] by Def6;
consider SA be ManySortedSet of Triangle A such that
A3:S.A = SA and
A4:for x be object st x in Triangle A holds
SA.x = [union rng (S|A).:([:L_x`1,{R_x}:]\/[:{L_x},L_R_x:]),
union rng(S|A).:([:R_L_x,{R_x}:]\/[:{L_x},R_R_x:])]
by A2,ORDINAL1:6;
set U =union rng (S|A);
A5:[x,y] in Triangle A by Def5;
[x,y]`1 =x & [x,y]`2=y;
then A6: x+y = [U.:([:L_x,{y}:]\/[:{x},L_y:] ),
U.:([:R_x,{y}:]\/[:{x},R_y:] )] by A1,A3,A5,A4;
A7: U.:[:L_x,{y}:] c= L_x ++ {y}
proof
let a be object;
assume a in U.:[:L_x,{y}:];
then consider b be object such that
A8: b in dom U & b in [:L_x,{y}:] & U.b = a by FUNCT_1:def 6;
consider x1,y1 be object such that
A9: x1 in L_x & y1 in {y} & b=[x1,y1] by ZFMISC_1:def 2,A8;
reconsider x1,y1 as Surreal by A9,SURREAL0:def 16;
set X1 = born x1,C = X1 (+) born y;
x1 in L_x\/ R_x by A9,XBOOLE_0:def 3;
then A10: C in A by SURREALO:1,ORDINAL7:94;
then A11: C in succ A by ORDINAL1:8;
then consider SC be ManySortedSet of Triangle C such that
A12:S.C = SC and
for x be object st x in Triangle C holds
SC.x = [union rng (S|C).:([:L_L_x,{R_x}:]\/[:{L_x},L_R_x:] ),
union rng(S|C).:([:R_L_x,{R_x}:]\/[:{L_x},R_R_x:] )] by A2;
A13:y=y1 by A9,TARSKI:def 1;
A14:dom SC = Triangle C by PARTFUN1:def 2;
A15: [x1,y] in Triangle C by Def5;
A16: x1 + y = SC.[x1,y] by A11,A1,A2,A12,Th26;
A17: SC. [x1,y] = (union rng S).[x1,y] by A11,A14,A15,Th2,A1,A12;
U.b = (union rng S).b by A12,Th5,A10,A13,A9,A15,A14;
hence thesis by A9,Def8,A8, A17,A16,A13;
end;
L_x ++ {y} c= U.:[:L_x,{y}:]
proof
let b be object;
assume b in L_x ++ {y};
then consider x1,y1 be Surreal such that
A18: x1 in L_x & y1 in {y} & b = x1 + y1 by Def8;
A19: y=y1 by A18,TARSKI:def 1;
set X1 = born x1,C = X1 (+) born y;
x1 in L_x\/ R_x by A18,XBOOLE_0:def 3;
then A20: C in A by SURREALO:1,ORDINAL7:94;
then A21: C in succ A by ORDINAL1:8;
then consider SC be ManySortedSet of Triangle C such that
A22:S.C = SC and
for x be object st x in Triangle C holds
SC.x = [union rng (S|C).:([:L_L_x,{R_x}:]\/[:{L_x},L_R_x:] ),
union rng(S|C).:([:R_L_x,{R_x}:]\/[:{L_x},R_R_x:] )] by A2;
A23:dom SC = Triangle C by PARTFUN1:def 2;
A24: [x1,y] in Triangle C by Def5;
A25: x1 + y = SC.[x1,y] by A21,A1,A2,A22,Th26;
SC. [x1,y] = (union rng S).[x1,y]
by A21,A23,A24,Th2,A1,A22;
then A26:U.[x1,y] = b by A25,A18,A19,A22,Th5,A20,A24,A23;
A27: [x1,y] in dom U by A22,Th5,A20,A24,A23;
[x1,y] in [:L_x,{y}:] by A18,A19,ZFMISC_1:87;
hence thesis by A26,A27,FUNCT_1:def 6;
end;
then A28:U.:[:L_x,{y}:] = L_x ++ {y} by A7,XBOOLE_0:def 10;
A29: U.:[:R_x,{y}:] c= R_x ++ {y}
proof
let a be object;
assume a in U.:[:R_x,{y}:];
then consider b be object such that
A30: b in dom U & b in [:R_x,{y}:] & U.b = a by FUNCT_1:def 6;
consider x1,y1 be object such that
A31: x1 in R_x & y1 in {y} & b=[x1,y1] by ZFMISC_1:def 2,A30;
reconsider x1,y1 as Surreal by A31,SURREAL0:def 16;
set X1 = born x1,C = X1 (+) born y;
x1 in L_x\/ R_x by A31,XBOOLE_0:def 3;
then A32: C in A by SURREALO:1,ORDINAL7:94;
then A33: C in succ A by ORDINAL1:8;
then consider SC be ManySortedSet of Triangle C such that
A34:S.C = SC and
for x be object st x in Triangle C holds
SC.x = [union rng (S|C).:([:L_L_x,{R_x}:]\/[:{L_x},L_R_x:] ),
union rng(S|C).:([:R_L_x,{R_x}:]\/[:{L_x},R_R_x:] )] by A2;
A35:y=y1 by A31,TARSKI:def 1;
A36:dom SC = Triangle C by PARTFUN1:def 2;
A37: [x1,y] in Triangle C by Def5;
A38: x1 + y = SC.[x1,y] by A33,A1,A2,A34,Th26;
A39: SC. [x1,y] = (union rng S).[x1,y] by A33,A36,A37,Th2,A1,A34;
U.b = (union rng S).b by A34,Th5,A32,A35,A31,A37,A36;
hence thesis by A31,Def8,A30,A39,A38,A35;
end;
R_x ++ {y} c= U.:[:R_x,{y}:]
proof
let b be object;
assume b in R_x ++ {y};
then consider x1,y1 be Surreal such that
A40: x1 in R_x & y1 in {y} & b = x1 + y1 by Def8;
A41: y=y1 by A40,TARSKI:def 1;
set X1 = born x1,C = X1 (+) born y;
x1 in L_x\/ R_x by A40,XBOOLE_0:def 3;
then A42: C in A by SURREALO:1,ORDINAL7:94;
then A43: C in succ A by ORDINAL1:8;
then consider SC be ManySortedSet of Triangle C such that
A44:S.C = SC and
for x be object st x in Triangle C holds
SC.x = [union rng (S|C).:([:L_L_x,{R_x}:]\/[:{L_x},L_R_x:] ),
union rng(S|C).:([:R_L_x,{R_x}:]\/[:{L_x},R_R_x:] )] by A2;
A45:dom SC = Triangle C by PARTFUN1:def 2;
A46: [x1,y] in Triangle C by Def5;
A47: x1 + y = SC.[x1,y] by A43,A1,A2,A44,Th26;
SC. [x1,y] = (union rng S).[x1,y]
by A43,A45,A46,Th2,A1,A44;
then A48:U.[x1,y] = b by A47,A44,Th5,A42,A46,A45,A40,A41;
A49: [x1,y] in dom U by A44,Th5,A42,A46,A45;
[x1,y] in [:R_x,{y}:] by A40,A41,ZFMISC_1:87;
hence thesis by A48,A49,FUNCT_1:def 6;
end;
then A50:U.:[:R_x,{y}:] = R_x ++ {y} by A29,XBOOLE_0:def 10;
A51: U.:[:{x},L_y:] c= {x} ++ L_y
proof
let a be object;
assume a in U.:[:{x},L_y:];
then consider b be object such that
A52: b in dom U & b in [:{x},L_y:] & U.b = a by FUNCT_1:def 6;
consider x1,y1 be object such that
A53: x1 in {x} & y1 in L_y & b=[x1,y1] by ZFMISC_1:def 2,A52;
reconsider x1,y1 as Surreal by A53,SURREAL0:def 16;
set Y1 = born y1,C = born x (+) Y1;
y1 in L_y\/ R_y by A53,XBOOLE_0:def 3;
then A54: C in A by SURREALO:1,ORDINAL7:94;
then A55: C in succ A by ORDINAL1:8;
then consider SC be ManySortedSet of Triangle C such that
A56:S.C = SC and
for x be object st x in Triangle C holds
SC.x = [union rng (S|C).:([:L_L_x,{R_x}:]\/[:{L_x},L_R_x:] ),
union rng(S|C).:([:R_L_x,{R_x}:]\/[:{L_x},R_R_x:] )] by A2;
A57:x=x1 by A53,TARSKI:def 1;
A58:dom SC = Triangle C by PARTFUN1:def 2;
A59: [x,y1] in Triangle C by Def5;
A60: x+ y1 = SC.[x,y1] by A55,A1,A2,A56,Th26;
A61: SC. [x,y1] = (union rng S).[x,y1] by A55,A58,A59,Th2,A1,A56;
U.b = (union rng S).b by A56,Th5,A54,A57,A53,A59,A58;
hence thesis by A53,Def8,A52, A61,A60,A57;
end;
{x} ++ L_y c= U.:[:{x},L_y:]
proof
let b be object;
assume b in {x} ++ L_y;
then consider x1,y1 be Surreal such that
A62: x1 in {x} & y1 in L_y & b = x1 + y1 by Def8;
A63: x=x1 by A62,TARSKI:def 1;
set Y1 = born y1,C = born x (+) Y1;
y1 in L_y\/ R_y by A62,XBOOLE_0:def 3;
then A64: C in A by SURREALO:1,ORDINAL7:94;
then A65: C in succ A by ORDINAL1:8;
then consider SC be ManySortedSet of Triangle C such that
A66:S.C = SC and
for x be object st x in Triangle C holds
SC.x = [union rng (S|C).:([:L_L_x,{R_x}:]\/[:{L_x},L_R_x:] ),
union rng(S|C).:([:R_L_x,{R_x}:]\/[:{L_x},R_R_x:] )] by A2;
A67:dom SC = Triangle C by PARTFUN1:def 2;
A68: [x,y1] in Triangle C by Def5;
A69: x + y1 = SC.[x,y1] by A65,A1,A2,A66,Th26;
SC. [x,y1] = (union rng S).[x,y1] by A65,A67,A68,Th2,A1,A66;
then A70:U.[x,y1] = b by A69,A66,Th5,A64,A68,A67,A62,A63;
A71: [x,y1] in dom U by A66,Th5,A64,A68,A67;
[x,y1] in [:{x},L_y:] by A62,A63,ZFMISC_1:87;
hence thesis by A70,A71,FUNCT_1:def 6;
end;
then A72:{x} ++ L_y = U.:[:{x},L_y:] by A51,XBOOLE_0:def 10;
A73: U.:[:{x},R_y:] c= {x} ++ R_y
proof
let a be object;
assume a in U.:[:{x},R_y:];
then consider b be object such that
A74: b in dom U & b in [:{x},R_y:] & U.b = a by FUNCT_1:def 6;
consider x1,y1 be object such that
A75: x1 in {x} & y1 in R_y & b=[x1,y1] by ZFMISC_1:def 2,A74;
reconsider x1,y1 as Surreal by A75,SURREAL0:def 16;
set Y1 = born y1,C = born x (+) Y1;
y1 in L_y\/ R_y by A75,XBOOLE_0:def 3;
then A76: C in A by SURREALO:1,ORDINAL7:94;
then A77: C in succ A by ORDINAL1:8;
then consider SC be ManySortedSet of Triangle C such that
A78:S.C = SC and
for x be object st x in Triangle C holds
SC.x = [union rng (S|C).:([:L_L_x,{R_x}:]\/[:{L_x},L_R_x:] ),
union rng(S|C).:([:R_L_x,{R_x}:]\/[:{L_x},R_R_x:] )] by A2;
A79:x=x1 by A75,TARSKI:def 1;
A80:dom SC = Triangle C by PARTFUN1:def 2;
A81: [x,y1] in Triangle C by Def5;
A82: x+ y1 = SC.[x,y1] by A77,A1,A2,A78,Th26;
A83: SC. [x,y1] = (union rng S).[x,y1] by A77,A80,A81,Th2,A1,A78;
U.b = (union rng S).b by A78,Th5,A76,A79,A75,A81,A80;
hence thesis by A75,Def8,A74, A83,A82,A79;
end;
{x} ++ R_y c= U.:[:{x},R_y:]
proof
let b be object;
assume b in {x} ++ R_y;
then consider x1,y1 be Surreal such that
A84: x1 in {x} & y1 in R_y & b = x1 + y1 by Def8;
A85: x=x1 by A84,TARSKI:def 1;
set Y1 = born y1,C = born x (+) Y1;
y1 in L_y\/ R_y by A84,XBOOLE_0:def 3;
then A86: C in A by SURREALO:1,ORDINAL7:94;
then A87: C in succ A by ORDINAL1:8;
then consider SC be ManySortedSet of Triangle C such that
A88:S.C = SC and
for x be object st x in Triangle C holds
SC.x = [union rng (S|C).:([:L_L_x,{R_x}:]\/[:{L_x},L_R_x:] ),
union rng(S|C).:([:R_L_x,{R_x}:]\/[:{L_x},R_R_x:] )] by A2;
A89:dom SC = Triangle C by PARTFUN1:def 2;
A90: [x,y1] in Triangle C by Def5;
A91: x + y1 = SC.[x,y1] by A87,A1,A2,A88,Th26;
SC. [x,y1] = (union rng S).[x,y1]
by A87,A89,A90,Th2,A1,A88;
then A92:U.[x,y1] = b by A91,A88,Th5,A86,A90,A89,A84,A85;
A93: [x,y1] in dom U by A88,Th5,A86,A90,A89;
[x,y1] in [:{x},R_y:] by A84,A85,ZFMISC_1:87;
hence thesis by A92,A93,FUNCT_1:def 6;
end;
then A94:{x} ++ R_y = U.:[:{x},R_y:] by A73,XBOOLE_0:def 10;
U.:([:L_x,{y}:]\/[:{x},L_y:]) = (L_x ++ {y}) \/({x} ++(L_y))
by A28,A72,RELAT_1:120;
hence thesis by A6,A50,A94,RELAT_1:120;
end;
::$N Commutativity of Addition for Surreal Number, Conway Ch.1 Th.3 (ii)
theorem Th29:
x + y = y + x
proof
defpred P[Ordinal] means
for x,y be Surreal st born x (+) born y c= $1 holds x + y = y + x;
A1: for D be Ordinal st for C be Ordinal st C in D holds P[C] holds P[D]
proof
let D be Ordinal such that A2: for C be Ordinal st C in D holds P[C];
A3: for x,y be Surreal st born x (+) born y c= D
for X,Y be surreal-membered set
st (X c= L_x\/R_x & Y={y}) or (Y c= L_y\/R_y & X={x})
holds X ++ Y c= Y ++ X
proof
let x,y be Surreal such that A4: born x (+) born y c= D;
let X,Y be surreal-membered set such that
A5:(X c= L_x\/R_x & Y={y}) or (Y c= L_y\/R_y & X={x});
let a be object;
assume a in X++Y;
then consider x1,y1 be Surreal such that
A6: x1 in X & y1 in Y & a=x1+y1 by Def8;
(born x1 in born x & born y1 = born y) or
(born x1=born x & born y1 in born y) by SURREALO:1,A5,A6,TARSKI:def 1;
then born x1 (+) born y1 in born x(+) born y by ORDINAL7:94;
then x1+y1 = y1+x1 by A2,A4;
hence thesis by A6,Def8;
end;
let x,y be Surreal such that A7: born x (+) born y c= D;
L_x c= L_x\/R_x by XBOOLE_1:7;
then L_x ++ {y} c= {y}++L_x c= L_x ++ {y} by A7,A3;
then A8: L_x ++ {y} = {y}++L_x by XBOOLE_0:def 10;
R_x c= L_x\/R_x by XBOOLE_1:7;
then R_x ++ {y} c= {y}++R_x c= R_x ++ {y} by A7,A3;
then A9: R_x ++ {y} = {y}++R_x by XBOOLE_0:def 10;
L_y c= L_y\/R_y by XBOOLE_1:7;
then L_y ++ {x} c= {x}++L_y c= L_y ++ {x} by A7,A3;
then A10: L_y ++ {x} = {x}++L_y by XBOOLE_0:def 10;
R_y c= L_y\/R_y by XBOOLE_1:7;
then A11:R_y ++ {x} c= {x}++R_y c= R_y ++ {x} by A7,A3;
thus x + y = [(L_x ++ {y})\/({x}++L_y), (R_x ++ {y}) \/({x}++R_y)] by Th28
.= [({y} ++ L_x)\/(L_y++{x}), ({y}++R_x) \/(R_y++{x})]
by A8,A9,A10,A11,XBOOLE_0:def 10
.= y+x by Th28;
end;
for D be Ordinal holds P[D] from ORDINAL1:sch 2(A1);
hence thesis;
end;
definition
let x,y be Surreal;
redefine func x + y; :: redefines + (SURREALR, definition)
commutativity by Th29;
end;
Lm4: for X,Y be set holds X++Y c= Y++X
proof
let X,Y be set;
let z be object;
assume z in X++Y;
then consider x,y be Surreal such that
A1: x in X & y in Y & z=x+y by Def8;
thus thesis by A1,Def8;
end;
theorem Th30:
for X,Y be set holds X++Y = Y++X
proof
let X,Y be set;
X++Y c= Y++X c= X ++ Y by Lm4;
hence thesis by XBOOLE_0:def 10;
end;
definition
let X,Y be set;
redefine func X ++ Y; :: redefines ++ (SURREALR, definition)
commutativity by Th30;
end;
Lm5: for D be Ordinal holds
(for x,y be Surreal st born x (+) born y c= D holds
x+y is surreal & x+y in Day (born x (+) born y))
& for x,y,z,xz,yz be Surreal st
(born x (+) born z) c= D &
(born y (+) born z) c= D & xz=x+z & yz = y+z holds xz <= yz iff x <= y
proof
defpred P[Ordinal] means
(for x,y be Surreal st born x (+) born y c= $1 holds
x+y is surreal & x+y in Day (born x (+) born y))
& for x,y,z,xz,yz be Surreal st (born x (+) born z) c= $1 &
(born y (+) born z) c= $1 & xz=x+z & yz = y+z holds xz <= yz iff x <= y;
A1: for D be Ordinal st for C be Ordinal st C in D holds P[C] holds P[D]
proof
let D be Ordinal such that A2: for C be Ordinal st C in D holds P[C];
thus for x,y be Surreal st born x (+) born y c= D holds
x+y is surreal & x+y in Day (born x (+) born y)
proof
let x,y be Surreal such that A3: born x (+) born y c= D;
set Bx=born x,By=born y, B=Bx(+)By;
A4: x in Day Bx & y in Day By by SURREAL0:def 18;
x=[L_x,R_x] & y=[L_y,R_y];
then A5: L_x << R_x & L_y << R_y by SURREAL0:46,A4;
A6:L_x ++ {y} << R_x ++ {y}
proof
let l,r be Surreal such that A7: l in L_x ++ {y} & r in R_x ++ {y};
consider l1,l2 be Surreal such that
A8: l1 in L_x & l2 in {y} & l = l1+l2 by A7,Def8;
consider r1,r2 be Surreal such that
A9: r1 in R_x & r2 in {y} & r = r1+r2 by A7,Def8;
A10:l2=y=r2 by A8,A9,TARSKI:def 1;
set bl = born l1, br =born r1,bb = bl\/br,B=bb (+) born y;
l1 in L_x \/R_x & r1 in L_x \/R_x by A8,A9,XBOOLE_0:def 3;
then bl in born x & br in born x by SURREALO:1;
then bb in born x by ORDINAL3:12;
then A11:B in born x (+) born y by ORDINAL7:94;
A12:bl (+)born y c= B by ORDINAL7:95,XBOOLE_1:7;
A13: born r1 (+) born y c= B by ORDINAL7:95,XBOOLE_1:7;
l1 < r1 by A8,A9,A5;
hence l < r by A3,A8,A9,A10,A11,A12,A13,A2;
end;
A14:L_x << {x} << R_x & L_y << {y} << R_y by SURREALO:11;
A15: {x}++L_y << R_x ++ {y}
proof
let l,r be Surreal such that A16: l in {x} ++ L_y & r in R_x ++ {y};
consider l1,l2 be Surreal such that
A17: l1 in {x} & l2 in L_y & l = l1+l2 by A16,Def8;
consider r1,r2 be Surreal such that
A18: r1 in R_x & r2 in {y} & r = r1+r2 by A16,Def8;
A19: l1=x & r2=y by A17,A18,TARSKI:def 1;
A20: l2 in L_y \/R_y & r1 in L_x \/R_x by A17,A18,XBOOLE_0:def 3;
then born r1(+)born l2 in born x(+)born l2 in born x(+)born y
by SURREALO:1,ORDINAL7:94;
then A21: born r1(+)born l2 in D by A3,ORDINAL1:10;
then reconsider rl=r1+l2,lr = l2+r1 as Surreal by A2;
born l1 (+) born l2 in born x(+)born y
by A20,SURREALO:1,A19,ORDINAL7:94;
then A22:born l1 (+) born l2 in D by A3;
set B1 = (born r1(+)born l2) \/ (born l1 (+) born l2);
A23: B1 in D by A21,A22,ORDINAL3:12;
A24: born l1 (+) born l2 c= B1 by XBOOLE_1:7;
A25: born r1(+)born l2 c= B1 by XBOOLE_1:7;
l1 < r1 by A18,A14,A17;
then A26: l < rl by A17,A24,A25,A23,A2;
born r1 (+) born r2 in born x(+)born y
by A20,SURREALO:1,A19,ORDINAL7:94;
then A27:born r1 (+) born r2 in D by A3;
set B2 = (born r1(+)born l2) \/ (born r1 (+) born r2);
A28: B2 in D by A21,A27,ORDINAL3:12;
A29: born r1 (+) born r2 c= B2 by XBOOLE_1:7;
A30: born r1(+)born l2 c= B2 by XBOOLE_1:7;
A31: l2 < r2 by A18,A14,A17;
assume r <= l;
then r <= rl by A26,SURREALO:4;
hence thesis by A18,A29,A30,A28,A2,A31;
end;
A32: L_x++{y} << {x} ++ R_y
proof
let l,r be Surreal such that A33: l in L_x++{y} & r in {x} ++ R_y;
consider l1,l2 be Surreal such that
A34: l1 in L_x & l2 in {y} & l = l1+l2 by A33,Def8;
consider r1,r2 be Surreal such that
A35: r1 in {x} & r2 in R_y & r = r1+r2 by A33,Def8;
A36: l2=y & r1=x by A34,A35,TARSKI:def 1;
A37: r2 in L_y \/R_y & l1 in L_x \/R_x by A34,A35,XBOOLE_0:def 3;
then born l1(+)born r2 in born x(+)born r2 in born x(+)born y
by SURREALO:1,ORDINAL7:94;
then A38: born l1(+)born r2 in D by A3,ORDINAL1:10;
then reconsider lr=l1+r2,rl = r2+l1 as Surreal by A2;
born r1 (+) born r2 in born x(+)born y
by A37,SURREALO:1,A36,ORDINAL7:94;
then A39:born r1 (+) born r2 in D by A3;
set B1 = (born l1(+)born r2) \/ (born r1 (+) born r2);
A40: B1 in D by A38,A39,ORDINAL3:12;
A41: born r1 (+) born r2 c= B1 by XBOOLE_1:7;
A42: born l1(+)born r2 c= B1 by XBOOLE_1:7;
l1 <r1 by A35,A14,A34;
then A43: lr < r by A35,A41,A42,A40,A2;
born l1 (+) born l2 in born x(+)born y
by A37,SURREALO:1,A36,ORDINAL7:94;
then A44:born l1 (+) born l2 in D by A3;
set B2 = (born l1(+)born r2) \/ (born l1 (+) born l2);
A45: B2 in D by A38,A44,ORDINAL3:12;
A46: born l1 (+) born l2 c= B2 by XBOOLE_1:7;
A47: born l1(+)born r2 c= B2 by XBOOLE_1:7;
A48: l2 < r2 by A35,A14,A34;
assume r <= l;
then lr <= l by A43,SURREALO:4;
hence thesis by A34,A46,A47,A45,A2,A48;
end;
A49:{x} ++ L_y << {x} ++ R_y
proof
let l,r be Surreal such that A50: l in {x}++ L_y & r in {x} ++ R_y;
consider l1,l2 be Surreal such that
A51: l1 in {x} & l2 in L_y & l = l1+l2 by A50,Def8;
consider r1,r2 be Surreal such that
A52: r1 in {x} & r2 in R_y & r = r1+r2 by A50,Def8;
A53: l1=x=r1 by A51,A52,TARSKI:def 1;
set bl = born l2, br =born r2,bb = bl\/br,B=bb (+) born x;
l2 in L_y \/R_y & r2 in L_y \/R_y by A51,A52,XBOOLE_0:def 3;
then bl in born y & br in born y by SURREALO:1;
then bb in born y by ORDINAL3:12;
then A54:B in born y (+) born x by ORDINAL7:94;
A55:bl (+)born x c= B by ORDINAL7:95,XBOOLE_1:7;
A56: br (+) born x c= B by ORDINAL7:95,XBOOLE_1:7;
l2 < r2 by A51,A52,A5;
hence l < r by A51,A52,A53,A54,A3,A55,A56,A2;
end;
A57: (L_x ++ {y})\/({x}++L_y) << (R_x ++ {y}) \/({x}++R_y)
proof
let l,r be Surreal;
assume l in (L_x ++ {y})\/({x}++L_y) & r in (R_x ++ {y}) \/({x}++R_y);
then (l in (L_x ++ {y}) or l in ({x}++L_y)) &
(r in (R_x ++ {y}) or r in ({x}++R_y)) by XBOOLE_0:def 3;
hence thesis by A6,A15,A32,A49;
end;
for z be object st
z in (L_x ++ {y})\/({x}++L_y) \/ ((R_x ++ {y}) \/({x}++R_y))
ex O be Ordinal st O in born x (+) born y & z in Day O
proof
let z be object;
assume z in (L_x ++ {y})\/({x}++L_y) \/ ((R_x ++ {y}) \/({x}++R_y));
then z in (L_x++{y})\/({x}++L_y) or z in (R_x++{y})\/({x}++R_y)
by XBOOLE_0:def 3;
then per cases by XBOOLE_0:def 3;
suppose z in L_x ++ {y} or z in R_x ++ {y};
then (ex u,w be Surreal st u in L_x & w in {y} & z=u+w) or
(ex u,w be Surreal st u in R_x & w in {y} & z=u+w) by Def8;
then consider u,w be Surreal such that
A58: (u in L_x or u in R_x )& w in {y} & z=u+w;
u in L_x \/R_x by A58,XBOOLE_0:def 3;
then born u in born x & w=y by A58,SURREALO:1,TARSKI:def 1;
then A59: born u (+) born w in born x (+) born y by ORDINAL7:94;
then P[born u (+) born w] by A3,A2;
hence thesis by A59,A58;
end;
suppose z in {x} ++ L_y or z in {x} ++ R_y;
then (ex u,w be Surreal st u in {x} & w in L_y & z=u+w) or
(ex u,w be Surreal st u in {x} & w in R_y & z=u+w) by Def8;
then consider u,w be Surreal such that
A60: u in {x} & (w in L_y or w in R_y) & z=u+w;
w in L_y \/R_y by A60,XBOOLE_0:def 3;
then born w in born y & u=x by A60,SURREALO:1,TARSKI:def 1;
then A61: born u (+) born w in born x (+) born y by ORDINAL7:94;
then P[born u (+) born w] by A3,A2;
hence thesis by A61,A60;
end;
end;
then [(L_x ++ {y})\/({x}++L_y), (R_x ++ {y}) \/({x}++R_y)]
in Day (born x (+) born y) by A57,SURREAL0:46;
hence thesis by Th28;
end;
defpred O[Ordinal] means
for x,y,z,xz,yz be Surreal st born x (+) born y (+) born z c= $1 &
(born x (+) born z) c= D & (born y (+) born z) c= D & xz=x+z & yz = y+z
holds xz <= yz iff x <= y;
A62: for E be Ordinal st for C be Ordinal st C in E holds O[C] holds O[E]
proof
let E be Ordinal;
assume A63:for C be Ordinal st C in E holds O[C];
let x,y,z,xz,yz be Surreal such that
A64: born x (+) born y (+) born z c= E &
(born x (+) born z) c= D &
(born y (+) born z) c= D & xz=x+z & yz = y+z;
A65:xz=[(L_x ++ {z})\/({x}++L_z),(R_x ++ {z})\/({x}++R_z)] by A64,Th28;
A66:yz=[(L_y ++ {z})\/({y}++L_z),(R_y ++ {z})\/({y}++R_z)] by A64,Th28;
thus xz <= yz implies x <= y
proof
assume xz <= yz;
then A67: L_xz << {yz} & {xz} << R_yz by SURREAL0:43;
A68: L_x << {y}
proof
let xL,r be Surreal such that A69:xL in L_x & r in {y};
A70:xL in L_x \/R_x by A69,XBOOLE_0:def 3;
then A71: born xL (+) born z in born x (+) born z
by SURREALO:1,ORDINAL7:94;
then reconsider xLz=xL+z as Surreal by A2,A64;
born xL (+) born y in born x (+) born y
by A70,SURREALO:1,ORDINAL7:94;
then A72: born xL (+) born y (+) born z in
born x (+) born y (+) born z by ORDINAL7:94;
A73: born xL (+) born z c= D by A71,A64,ORDINAL1:def 2;
A74:r=y by A69,TARSKI:def 1;
assume not xL < r;
then A75:yz <= xLz by A74,A72,A64,A63,A73;
z in {z} by TARSKI:def 1;
then xLz in L_x ++ {z} by A69,Def8;
then A76: xLz in L_xz by A65,XBOOLE_0:def 3;
yz in {yz} by TARSKI:def 1;
hence thesis by A75,A76,A67;
end;
{x} << R_y
proof
let l,yR be Surreal such that A77:l in {x} & yR in R_y;
A78: yR in L_y \/R_y by A77,XBOOLE_0:def 3;
then A79: born yR (+) born z in born y (+) born z
by SURREALO:1,ORDINAL7:94;
then reconsider yRz=yR+z as Surreal by A64,A2;
born x (+) born yR in born x (+) born y
by A78,SURREALO:1,ORDINAL7:94;
then A80: born x (+) born yR (+) born z in
born x (+) born y (+) born z by ORDINAL7:94;
A81: born yR (+) born z c= D by A79,A64,ORDINAL1:def 2;
A82:l=x by A77,TARSKI:def 1;
assume not l < yR;
then A83:yRz <= xz by A82,A80,A63,A64,A81;
z in {z} by TARSKI:def 1;
then yRz in R_y ++ {z} by A77,Def8;
then A84: yRz in R_yz by A66,XBOOLE_0:def 3;
xz in {xz} by TARSKI:def 1;
hence thesis by A83,A84,A67;
end;
hence thesis by A68,SURREAL0:43;
end;
assume A85: x <=y;
then A86:L_x << {y} & {x} << R_y by SURREAL0:43;
A87: L_x ++ {z} <<{yz}
proof
let L,R be Surreal such that A88:L in L_x ++ {z} & R in {yz};
consider x1,z1 be Surreal such that
A89: x1 in L_x & z1 in {z} & L = x1+z1 by A88,Def8;
A90:z1=z & R=yz by A88,A89,TARSKI:def 1;
A91: x1 in L_x\/R_x by A89,XBOOLE_0:def 3;
then born x1 (+) born y in born x (+) born y
by SURREALO:1,ORDINAL7:94;
then A92: born x1 (+) born y (+) born z in
born x (+) born y (+) born z by ORDINAL7:94;
y in {y} by TARSKI:def 1;
then A93: x1 < y by A89,A86;
born x1 (+) born z in born x (+) born z by A91,SURREALO:1,ORDINAL7:94;
then (born x1 (+) born z) c= D by A64,ORDINAL1:def 2;
hence L <R by A93,A90,A64,A92,A63,A89;
end;
A94: {x}++L_z <<{yz}
proof
let L,R be Surreal such that A95:L in {x} ++ L_z & R in {yz};
consider x1,z1 be Surreal such that
A96: x1 in {x} & z1 in L_z & L = x1+z1 by A95,Def8;
A97:x1=x & R=yz by A95,A96,TARSKI:def 1;
assume A98: R <= L;
A99: z1 in L_z \/R_z by A96,XBOOLE_0:def 3;
then A100: born x (+) born z1 in born x (+) born z &
born y (+) born z1 in born y (+) born z by SURREALO:1,ORDINAL7:94;
then reconsider xz1=x+z1,yz1=y+z1 as Surreal by A2,A64;
A101:born x (+) born y (+) born z1 in born x (+) born y (+) born z
by A99,SURREALO:1,ORDINAL7:94;
A102: born x (+) born z1 c= D & born y (+) born z1 c= D
by A100,A64,ORDINAL1:def 2;
A103: xz1 <= yz1 by A85,A101,A64,A63,A102;
A104:yz= [(L_y ++ {z})\/({y}++L_z),(R_y ++ {z})\/({y}++R_z)]
by Th28,A64;
A105: L_yz << {yz} by SURREALO:11;
A106: yz in {yz} by TARSKI:def 1;
y in {y} by TARSKI:def 1;
then yz1 in {y}++L_z by A96,Def8;
then yz1 in (L_y ++ {z})\/({y}++L_z) by XBOOLE_0:def 3;
hence thesis by A103,A106,A105,A104,A96,A97,A98,SURREALO:4;
end;
A107: {xz} << R_y ++ {z}
proof
let L,R be Surreal such that A108:L in {xz} & R in R_y ++ {z};
consider y1,z1 be Surreal such that
A109:y1 in R_y & z1 in {z} & R = y1+z1 by A108,Def8;
A110:z1=z & L=xz by A108,A109,TARSKI:def 1;
A111:y1 in L_y\/R_y by A109,XBOOLE_0:def 3;
then born x (+) born y1 in born x (+) born y
by SURREALO:1,ORDINAL7:94;
then A112: born x (+) born y1 (+) born z in
born x (+) born y (+) born z by ORDINAL7:94;
x in {x} by TARSKI:def 1;
then A113: x < y1 by A109,A86;
born y1 (+) born z in born y (+) born z
by A111,SURREALO:1,ORDINAL7:94;
then (born y1 (+) born z) c= D by A64,ORDINAL1:def 2;
hence L <R by A113,A110,A64,A112,A63,A109;
end;
A114:{xz} << {y}++R_z
proof
let L,R be Surreal such that A115:L in {xz} & R in {y}++R_z;
consider y1,z2 be Surreal such that
A116:y1 in {y} & z2 in R_z & R = y1+z2 by A115,Def8;
A117:y1=y & L=xz by A115,A116,TARSKI:def 1;
assume A118: R <= L;
A119: z2 in L_z \/R_z by A116,XBOOLE_0:def 3;
then A120: born x (+) born z2 in born x (+) born z &
born y (+) born z2 in born y (+) born z by SURREALO:1,ORDINAL7:94;
then reconsider xz2=x+z2,yz2=y+z2 as Surreal by A2,A64;
A121: born x (+) born y (+) born z2 in born x (+) born y (+) born z
by A119,SURREALO:1,ORDINAL7:94;
born x (+) born z2 c= D & born y (+) born z2 c= D
by A120,A64,ORDINAL1:def 2;
then
A122: xz2 <= yz2 by A64,A63,A121,A85;
A123:xz= [(L_x ++ {z})\/({x}++L_z), (R_x ++ {z}) \/({x}++R_z)]
by Th28,A64;
A124: {xz} << R_xz by SURREALO:11;
A125: xz in {xz} by TARSKI:def 1;
x in {x} by TARSKI:def 1;
then xz2 in {x}++R_z by A116,Def8;
then xz2 in (R_x ++ {z}) \/({x}++R_z) by XBOOLE_0:def 3;
hence thesis by A123,A122,A125,A124,A116,A117,A118,SURREALO:4;
end;
A126: L_xz << {yz}
proof
let L,R be Surreal such that A127:L in L_xz & R in {yz};
xz= [(L_x ++ {z})\/({x}++L_z), (R_x ++ {z}) \/({x}++R_z)] by Th28,A64;
then L in (L_x ++ {z}) or L in ({x}++L_z) by A127,XBOOLE_0:def 3;
hence thesis by A127,A87,A94;
end;
{xz} << R_yz
proof
let L,R be Surreal such that A128:L in {xz} & R in R_yz;
yz= [(L_y ++ {z})\/({y}++L_z), (R_y ++ {z}) \/({y}++R_z)] by Th28,A64;
then R in (R_y ++ {z}) or R in ({y}++R_z) by A128,XBOOLE_0:def 3;
hence thesis by A128,A107,A114;
end;
hence xz <= yz by A126,SURREAL0:43;
end;
A129:for E be Ordinal holds O[E] from ORDINAL1:sch 2(A62);
let x,y,z,xz,yz be Surreal such that
A130:(born x (+) born z) c= D &
(born y (+) born z) c= D & xz=x+z & yz = y+z;
O[born x (+) born y (+) born z] by A129;
hence thesis by A130;
end;
for E be Ordinal holds P[E] from ORDINAL1:sch 2(A1);
hence thesis;
end;
registration
let x,y;
cluster x + y -> surreal;
coherence by Lm5;
end;
definition
let x,y be Surreal;
func x - y -> Surreal equals
x + (-y);
coherence;
end;
theorem
born (x+y) c= born x (+) born y
proof
x+y in Day (born x (+) born y) by Lm5;
hence thesis by SURREAL0:def 18;
end;
registration
let X,Y be set;
cluster X ++ Y -> surreal-membered;
coherence
proof
let z be object;
assume z in X ++ Y;
then ex x,y be Surreal st x in X & y in Y & x+y=z by Def8;
hence thesis;
end;
end;
::$N Transitive Law of Addition for Surreal Number, Conway Ch.1 Th.5
theorem Th32:
x <= y iff x+z <= y+z
proof
set Bx=born x,By=born y,Bz=born z;
set Bxz=Bx(+)Bz,Byz=By(+)Bz;
Bxz c= Bxz\/Byz & Byz c= Bxz\/Byz by XBOOLE_1:7;
hence thesis by Lm5;
end;
theorem Th33:
for X1,X2,Y be set holds
(X1\/X2) ++ Y = (X1 ++ Y) \/ (X2++Y)
proof
let X1,X2,Y be set;
thus (X1\/X2) ++ Y c= (X1 ++ Y) \/ (X2++Y)
proof
let xy be object;
assume xy in (X1\/X2)++Y;
then consider x,y be Surreal such that
A1: x in X1\/X2 & y in Y & xy=x+y by Def8;
x in X1 or x in X2 by A1,XBOOLE_0:def 3;
then xy in (X1 ++ Y) or xy in (X2++Y) by A1,Def8;
hence thesis by XBOOLE_0:def 3;
end;
let xy be object;
assume xy in (X1 ++ Y) \/ (X2++Y);
then per cases by XBOOLE_0:def 3;
suppose xy in X1++Y;
then consider x,y be Surreal such that
A2: x in X1 & y in Y & xy=x+y by Def8;
x in X1\/X2 by A2,XBOOLE_0:def 3;
hence thesis by A2,Def8;
end;
suppose xy in X2++Y;
then consider x,y be Surreal such that
A3: x in X2 & y in Y & xy=x+y by Def8;
x in X1\/X2 by A3,XBOOLE_0:def 3;
hence thesis by A3,Def8;
end;
end;
theorem
for X,Y1,Y2 be set holds
X ++ (Y1\/Y2) = (X ++ Y1) \/ (X++Y2) by Th33;
theorem
for X1,X2,Y1,Y2 be set st X1 <=_ X2 & Y1 <=_ Y2
holds X1++Y1 <=_ X2++Y2
proof
let X1,X2,Y1,Y2 be set such that
A1: X1 <=_ X2 & Y1 <=_ Y2;
let x such that A2: x in X1++Y1;
consider x1,y1 be Surreal such that
A3:x1 in X1 & y1 in Y1 & x = x1 + y1 by A2,Def8;
consider x2,x3 be Surreal such that
A4:x2 in X2 & x3 in X2 & x2 <= x1 <= x3 by A3,A1;
consider y2,y3 be Surreal such that
A5:y2 in Y2 & y3 in Y2 & y2 <= y1 <= y3 by A3,A1;
take z2=x2+y2,z3=x3+y3;
x2+y2 <= x1+y2 <= x1 +y1 <= x3+y1 <= x3+y3 by A4,A5,Th32;
hence thesis by A4,A5,Def8,A3,SURREALO:4;
end;
theorem Th36:
{x}++{y}={x+y}
proof
thus {x}++{y}c={x+y}
proof
let xy be object;
assume xy in {x}++{y};
then consider a,b be Surreal such that
A1:a in {x} & b in {y} & xy=a+b by Def8;
a=x & b=y by A1,TARSKI:def 1;
hence thesis by A1,TARSKI:def 1;
end;
A2:x in {x} & y in {y} by TARSKI:def 1;
o in {x+y} implies o in {x}++{y}
proof
assume o in {x+y};
then o=x+y by TARSKI:def 1;
hence thesis by A2,Def8;
end;
hence thesis by TARSKI:def 3;
end;
::$N Associativity of Addition for Surreal Number, Conway Ch.1 Th.3(iii)
theorem Th37:
(x+y)+z = x+(y+z)
proof
defpred P[Ordinal] means
for x,y,z be Surreal st born x (+) born y (+) born z c= $1
holds (x+y)+z = x+(y+z);
A1:for D be Ordinal st for C be Ordinal st C in D holds P[C] holds P[D]
proof
let D be Ordinal such that A2: for C be Ordinal st C in D holds P[C];
A3:for X,Y,Z be set st for x,y,z be Surreal st x in X & y in Y & z in Z
holds born x (+) born y (+) born z in D holds
X++(Y++Z) = (X++Y)++Z
proof
let X,Y,Z be set such that
A4:for x,y,z be Surreal st x in X & y in Y & z in Z holds
born x (+) born y (+) born z in D;
thus X++(Y++Z) c= (X++Y)++Z
proof
let xyz be object;
assume xyz in X++(Y++Z);
then consider x,yz be Surreal such that
A5: x in X & yz in Y++Z & xyz = x+yz by Def8;
consider y,z be Surreal such that
A6: y in Y & z in Z & yz = y+z by A5,Def8;
x+y in X++Y by A5,A6,Def8;
then A7:x+y+z in (X++Y)++Z by A6,Def8;
born x (+) born y (+) born z in D by A5,A6,A4;
hence thesis by A5,A6,A7,A2;
end;
let xyz be object;
assume xyz in X++Y++Z;
then consider xy,z be Surreal such that
A8: xy in X++Y & z in Z & xyz = xy+z by Def8;
consider x,y be Surreal such that
A9: x in X & y in Y & xy = x+y by A8,Def8;
y+z in Y++Z by A8,A9,Def8;
then A10:x+(y+z) in X++(Y++Z) by A9,Def8;
born x (+) born y (+) born z in D by A8,A9,A4;
hence thesis by A8,A9,A10,A2;
end;
let x,y,z be Surreal such that A11:born x (+) born y (+) born z c= D;
set xy=x+y,yz=y+z;
A12: x + y=[(L_x ++ {y})\/({x} ++ L_y),(R_x ++ {y})\/({x} ++ R_y)]
by Th28;
A13: y + z=[(L_y ++ {z})\/({y} ++ L_z),(R_y ++ {z})\/({y} ++ R_z)]
by Th28;
A14:for a,b,c be Surreal st a in L_x & b in {y} & c in {z} holds
born a (+) born b (+) born c in D
proof
let a,b,c be Surreal;
assume a in L_x & b in {y} & c in {z};
then A15:a in L_x\/R_x & b =y & c = z by TARSKI:def 1,XBOOLE_0:def 3;
then born a (+) born b in born x (+) born y by SURREALO:1,ORDINAL7:94;
then born a (+) born b (+) born c in
born x (+) born y (+) born z by ORDINAL7:94,A15;
hence thesis by A11;
end;
A16:for a,b,c be Surreal st a in {x} & b in L_y & c in {z} holds
born a (+) born b (+) born c in D
proof
let a,b,c be Surreal;
assume a in {x} & b in L_y& c in {z};
then A17:a=x & b in L_y\/R_y & c = z by TARSKI:def 1,XBOOLE_0:def 3;
then born a (+) born b in born x (+) born y by SURREALO:1,ORDINAL7:94;
then born a (+) born b (+) born c in
born x (+) born y (+) born z by ORDINAL7:94,A17;
hence thesis by A11;
end;
A18:for a,b,c be Surreal st a in {x} & b in {y} & c in L_z holds
born a (+) born b (+) born c in D
proof
let a,b,c be Surreal;
assume a in {x} & b in {y} & c in L_z;
then a=x & b=y & c in L_z\/R_z by TARSKI:def 1,XBOOLE_0:def 3;
then born a (+) born b (+) born c in
born x (+) born y (+) born z by SURREALO:1,ORDINAL7:94;
hence thesis by A11;
end;
A19:(((L_x ++ {y})\/({x} ++ L_y)) ++ {z})\/({xy} ++ L_z)
= ((L_x ++ {y}) ++ {z})\/(({x} ++ L_y) ++ {z})\/({xy} ++ L_z) by Th33
.= (L_x ++ ({y} ++ {z}))\/(({x} ++ L_y) ++ {z})\/({xy} ++ L_z) by A14,A3
.= (L_x ++ ({y} ++ {z}))\/({x} ++ (L_y ++ {z}))\/({xy} ++ L_z) by A16,A3
.= (L_x ++ ({y} ++ {z}))\/({x} ++ (L_y ++ {z}))\/(( {x}++{y}) ++ L_z)
by Th36
.= (L_x ++ ({y} ++ {z}))\/({x} ++ (L_y ++ {z}))\/({x}++({y} ++ L_z))
by A18,A3
.= (L_x ++ ({y} ++ {z}))\/(({x} ++ (L_y ++ {z}))\/({x}++({y} ++ L_z)))
by XBOOLE_1:4
.= (L_x ++ ({y} ++ {z}))\/({x} ++ ((L_y ++ {z})\/({y} ++ L_z))) by Th33
.= (L_x ++ ({yz}))\/({x} ++ (L_yz)) by A13,Th36;
A20:for a,b,c be Surreal st a in R_x & b in {y} & c in {z} holds
born a (+) born b (+) born c in D
proof
let a,b,c be Surreal;
assume a in R_x & b in {y} & c in {z};
then A21:a in L_x\/R_x & b =y & c = z by TARSKI:def 1,XBOOLE_0:def 3;
then born a (+) born b in born x (+) born y by SURREALO:1,ORDINAL7:94;
then born a (+) born b (+) born c in
born x (+) born y (+) born z by ORDINAL7:94,A21;
hence thesis by A11;
end;
A22:for a,b,c be Surreal st a in {x} & b in R_y & c in {z} holds
born a (+) born b (+) born c in D
proof
let a,b,c be Surreal;
assume a in {x} & b in R_y& c in {z};
then A23:a=x & b in L_y\/R_y & c = z by TARSKI:def 1,XBOOLE_0:def 3;
then born a (+) born b in born x (+) born y by SURREALO:1,ORDINAL7:94;
then born a (+) born b (+) born c in
born x (+) born y (+) born z by ORDINAL7:94,A23;
hence thesis by A11;
end;
A24:for a,b,c be Surreal st a in {x} & b in {y} & c in R_z holds
born a (+) born b (+) born c in D
proof
let a,b,c be Surreal;
assume a in {x} & b in {y} & c in R_z;
then a=x & b=y & c in L_z\/R_z by TARSKI:def 1,XBOOLE_0:def 3;
then born a (+) born b (+) born c in
born x (+) born y (+) born z by SURREALO:1,ORDINAL7:94;
hence thesis by A11;
end;
A25: (((R_x ++ {y}) \/({x} ++ R_y)) ++ {z}) \/({xy} ++ R_z) =
((R_x ++ {y})++ {z}) \/(({x} ++ R_y) ++ {z})\/({xy} ++ R_z) by Th33
.= (R_x ++ ({y}++ {z})) \/(({x} ++ R_y) ++ {z})\/({xy} ++ R_z) by A20,A3
.= (R_x ++ ({yz})) \/(({x} ++ R_y) ++ {z})\/({xy} ++ R_z) by Th36
.= (R_x ++ {yz}) \/(({x} ++ R_y) ++ {z})\/(({x}++{y}) ++ R_z) by Th36
.= (R_x ++ {yz}) \/({x} ++ (R_y ++ {z}))\/(({x}++{y}) ++ R_z) by A22,A3
.= (R_x ++ {yz}) \/({x} ++ (R_y ++ {z}))\/({x}++({y} ++ R_z)) by A24,A3
.= (R_x ++ {yz}) \/(({x} ++ (R_y ++ {z}))\/({x}++({y} ++ R_z)))
by XBOOLE_1:4
.= (R_x ++ {yz}) \/({x} ++ R_yz) by A13,Th33;
xy +z = [(L_xy ++ {z})\/({xy} ++ L_z), (R_xy ++ {z}) \/({xy} ++ R_z)]
by Th28
.= [(L_x ++ ({yz}))\/({x} ++ (L_yz)),
(R_x ++ {yz}) \/({x} ++ R_yz)] by A25,A19,A12;
hence thesis by Th28;
end;
A26:for D be Ordinal holds P[D] from ORDINAL1:sch 2(A1);
P[born x (+) born y (+) born z] by A26;
hence thesis;
end;
::$N Additive Identity for Surreal Number, Conway Ch.1 Th.3(i)
theorem Th38:
x + 0_No = x
proof
set y=0_No;
defpred P[Ordinal] means
for x be Surreal st born x = $1 holds x + y = x;
A1: for D be Ordinal st for C be Ordinal st C in D holds P[C] holds P[D]
proof
let D be Ordinal such that A2: for C be Ordinal st C in D holds P[C];
let x be Surreal such that A3: born x=D;
{x} ++ L_y = {} & {x} ++ R_y = {} by Th27;
then A4:(L_x ++ {y})\/({x} ++ L_y) = L_x ++ {y} &
(R_x ++ {y}) \/({x} ++ R_y)=R_x ++ {y};
A5:for Y be set st Y c= L_x\/R_x holds Y ++{y}=Y
proof
let Y be set such that A6:Y c= L_x\/R_x;
thus Y ++{y}c=Y
proof
let xy be object such that
A7: xy in Y ++ {y};
consider x1,y1 be Surreal such that
A8: x1 in Y & y1 in {y} & xy=x1+y1 by A7,Def8;
y1=y by A8,TARSKI:def 1;
hence thesis by A2,A8,A6,SURREALO:1,A3;
end;
let xy be object;
assume A9:xy in Y;
then reconsider xy as Surreal by SURREAL0:def 16,A6;
A10:y in {y} by TARSKI:def 1;
xy + y =xy by A2,A9,A6,A3,SURREALO:1;
hence thesis by A10,A9,Def8;
end;
L_x ++ {y} = L_x & R_x ++ {y} = R_x by XBOOLE_1:7,A5;
then x+y = [L_x,R_x] by A4,Th28;
hence thesis;
end;
for D be Ordinal holds P[D] from ORDINAL1:sch 2(A1);
hence thesis;
end;
registration
let x;
reduce x + 0_No to x;
reducibility by Th38;
end;
::$N Property of The Aditive Inverse for Surreal Number, Conway Ch.1 Th.4(iii)
theorem Th39:
x - x == 0_No
proof
set y=0_No;
defpred P[Ordinal] means
for x be Surreal st born x = $1 holds x + -x == y;
A1: for D be Ordinal st for C be Ordinal st C in D holds P[C] holds P[D]
proof
let D be Ordinal such that A2: for C be Ordinal st C in D holds P[C];
let x be Surreal such that A3: born x=D;
set X=x+-x;
A4: -x = [-- R_x, --L_x] by Th7;
then L_(-x) = --R_x & R_(-x) = --L_x;
then A5:x +- x =
[(L_x ++ {-x})\/({x} ++ --R_x), (R_x ++ {-x}) \/({x} ++ --L_x)] by Th28;
A6:L_X << {y}
proof
let l,r be Surreal such that
A7: l in L_X & r in {y} & r <= l;
A8:r=y by A7,TARSKI:def 1;
A9:{r} << R_l by A7,SURREAL0:43;
per cases by A5,A7,XBOOLE_0:def 3;
suppose
l in {x} ++ --R_x;
then consider ll,lr be Surreal such that
A10: ll in {x} & lr in --R_x & l = ll+lr by Def8;
consider xr be Surreal such that
A11: xr in R_x & -xr = lr by A10,Def4;
x=ll by A10,TARSKI:def 1;
then A12: l = [(L_x ++ {-xr})\/({x} ++ L_(-xr)),
(R_x ++ {-xr}) \/({x} ++ R_(-xr))] by Th28,A10,A11;
xr in L_x \/R_x by A11,XBOOLE_0:def 3;
then A13: xr +- xr == y by A3,A2,SURREALO:1;
-xr in {-xr} by TARSKI:def 1;
then xr +-xr in (R_x ++ {-xr}) by A11,Def8;
then xr +-xr in R_l by A12,XBOOLE_0:def 3;
hence thesis by A13,A9,A7,A8;
end;
suppose
l in L_x ++ {-x};
then consider xl,lr be Surreal such that
A14: xl in L_x & lr in {-x} & l = xl+lr by Def8;
lr = -x by A14,TARSKI:def 1;
then A15:l = [(L_xl ++ {-x})\/({xl} ++ L_(-x)),
(R_xl ++ {-x}) \/({xl} ++ R_(-x))] by A14,Th28;
xl in L_x \/R_x by A14,XBOOLE_0:def 3;
then A16: xl +- xl == y by A3,A2,SURREALO:1;
-xl in --(L_x) & xl in {xl} by A14,Def4,TARSKI:def 1;
then xl +- xl in ({xl} ++ --(L_x)) by Def8;
then xl +- xl in R_l by A4,A15,XBOOLE_0:def 3;
hence contradiction by A16,A9,A7,A8;
end;
end;
{X} << R_y;
hence X <= y by A6,SURREAL0:43;
A17:{y} << R_X
proof
let l,r be Surreal such that
A18: l in {y} & r in R_X & r <= l;
A19:l=y by A18,TARSKI:def 1;
A20: L_r << {l} by SURREAL0:43,A18;
per cases by A5,A18,XBOOLE_0:def 3;
suppose r in R_x ++ {-x};
then consider ll,lr be Surreal such that
A21: ll in R_x & lr in {-x} & r = ll+lr by Def8;
lr = -x by A21,TARSKI:def 1;
then A22: r = [(L_ll ++ {-x})\/({ll} ++ L_(-x)),
(R_ll ++ {-x}) \/({ll} ++ R_(-x))] by A21,Th28;
ll in L_x \/R_x by A21,XBOOLE_0:def 3;
then A23: ll +- ll == y by A3,A2,SURREALO:1;
-ll in L_(-x) & ll in {ll} by A21,Def4,A4,TARSKI:def 1;
then ll +- ll in {ll} ++ L_(-x) by Def8;
then ll +- ll in L_r by A22,XBOOLE_0:def 3;
hence thesis by A23,A20,A19,A18;
end;
suppose r in {x} ++ --L_x;
then consider ll,lr be Surreal such that
A24: ll in {x} & lr in --L_x & r = ll+lr by Def8;
consider xr be Surreal such that
A25: xr in L_x & -xr = lr by A24,Def4;
x=ll by A24,TARSKI:def 1;
then A26:r = [(L_x ++ {-xr})\/({x} ++ L_(-xr)),
(R_x ++ {-xr}) \/({x} ++ R_(-xr))]
by Th28,A24,A25;
xr in L_x \/R_x by A25,XBOOLE_0:def 3;
then A27: xr +- xr == y by A3,A2,SURREALO:1;
-xr in {-xr} by TARSKI:def 1;
then xr +-xr in (L_x ++ {-xr}) by A25,Def8;
then xr +-xr in L_r by A26,XBOOLE_0:def 3;
hence thesis by A27,A20,A18,A19;
end;
end;
L_y << {X};
hence thesis by SURREAL0:43,A17;
end;
for D be Ordinal holds P[D] from ORDINAL1:sch 2(A1);
hence thesis;
end;
::$N Conway Ch.1 Th.4(i)
theorem Th40:
-(x+y) = -x + -y
proof
defpred P[Ordinal] means
for x,y be Surreal st born x (+) born y c= $1 holds - (x + y) = -x + -y;
A1: for D be Ordinal st for C be Ordinal st C in D holds P[C] holds P[D]
proof
let D be Ordinal such that A2: for C be Ordinal st C in D holds P[C];
A3: for x,y be Surreal st born x (+) born y c= D
for X,Y be surreal-membered set
st (X c= L_x\/R_x & Y={y}) or (Y c= L_y\/R_y & X={x})
holds -- (X ++ Y) = (--X) ++ (--Y)
proof
let x,y be Surreal such that A4: born x (+) born y c= D;
let X,Y be surreal-membered set such that
A5:(X c= L_x\/R_x & Y={y}) or (Y c= L_y\/R_y & X={x});
thus -- (X ++ Y) c= (--X) ++ (--Y)
proof
let a be object;
assume a in -- (X ++ Y);
then consider z be Surreal such that
A6:z in (X ++ Y) & a=-z by Def4;
consider x1,y1 be Surreal such that
A7: x1 in X & y1 in Y & z=x1+y1 by A6,Def8;
(born x1 in born x & born y1 = born y) or
(born x1=born x & born y1 in born y) by SURREALO:1,A5,A7,TARSKI:def 1;
then born x1 (+) born y1 in born x(+) born y by ORDINAL7:94;
then A8: a = (-x1) + (-y1) by A2,A4,A6,A7;
-x1 in --X & -y1 in --Y by A7,Def4;
hence thesis by A8,Def8;
end;
let a be object;
assume a in (--X) ++ (--Y);
then consider x2,y2 be Surreal such that
A9: x2 in --X & y2 in --Y & a=x2+y2 by Def8;
consider x1 be Surreal such that
A10: x1 in X & x2=-x1 by A9,Def4;
consider y1 be Surreal such that
A11: y1 in Y & y2=-y1 by A9,Def4;
A12: x1+y1 in X ++ Y by A10,A11,Def8;
(born x1 in born x & born y1 = born y) or
(born x1=born x & born y1 in born y)
by SURREALO:1,A5,A10,A11,TARSKI:def 1;
then born x1 (+) born y1 in born x(+) born y by ORDINAL7:94;
then -(x1+y1) = a by A2,A4,A10,A11,A9;
hence thesis by A12,Def4;
end;
let x,y be Surreal such that A13: born x (+) born y c= D;
A14:L_x c= L_x\/R_x by XBOOLE_1:7;
A15:R_x c= L_x\/R_x by XBOOLE_1:7;
A16:L_y c= L_y\/R_y by XBOOLE_1:7;
A17:R_y c= L_y\/R_y by XBOOLE_1:7;
A18: x + y = [(L_x ++ {y})\/({x}++L_y), (R_x ++ {y}) \/({x}++R_y)]
by Th28;
A19: L_(-x) = --R_x & R_(-x) = --L_x & (-y)`1 = --R_y & (-y)`2 = --L_y
by Th8;
A20:{-x} = --{x} & {-y} = --{y} by Th21;
A21: R_(-x) ++ {-y} = --(L_x ++{y}) by A19,A20, A14,A3,A13;
A22: {-x}++R_(-y) =--({x}++ L_y) by A19,A20,A16,A3,A13;
A23: L_(-x) ++ {-y} = --(R_x ++{y}) by A19,A20,A15,A3,A13;
{-x}++L_(-y) =--({x}++ R_y) by A19,A20,A17,A3,A13;
then A24: (L_(-x) ++ {-y})\/({-x}++L_(-y)) =
-- ((R_x ++{y})\/({x}++ R_y)) by A23,Th20;
thus -x + -y = [(L_(-x) ++ {-y})\/({-x}++L_(-y)),
(R_(-x) ++ {-y}) \/({-x}++R_(-y))] by Th28
.= [-- R_(x+y), -- L_(x+y)] by A18,A22,A21,Th20,A24
.= - (x+y) by Th7;
end;
for D be Ordinal holds P[D] from ORDINAL1:sch 2(A1);
hence thesis;
end;
theorem Th41:
x+y <= z iff x <= z - y
proof
thus x+y <= z implies x <= z - y
proof
assume x+y <= z;
then A1: x+y +-y <= z +-y by Th32;
y -y == 0_No by Th39;
then x+(y +-y) >= x+0_No by Th32;
then x+y +-y >= x by Th37;
hence thesis by A1,SURREALO:4;
end;
assume x <= z - y;
then x+y <= z +- y +y by Th32;
then A2: x+y <= z + (- y +y) by Th37;
y -y == 0_No by Th39;
then z + (-y +y) <=z+0_No =z by Th32;
hence thesis by A2,SURREALO:4;
end;
theorem Th42:
x+y < z iff x < z - y
proof
thus x+y < z implies x < z - y
proof
assume A1: x+y < z & not x < z - y;
then x+y >= (z +- y)+y by Th32;
then A2: x+y >= z +(- y+y) by Th37;
y -y == 0_No by Th39;
then z + (-y +y) >=z+0_No =z by Th32;
hence thesis by A2,A1,SURREALO:4;
end;
assume A3:x < z - y & z <= x+y;
then z +-y <= x+y+-y by Th32;
then A4:z +-y <= x+(y+-y) by Th37;
y -y == 0_No by Th39;
then x+(y+-y) <=x+0_No =x by Th32;
hence thesis by A4,A3,SURREALO:4;
end;
theorem Th43:
x <= y & z <= t implies x+z <= y+t
proof
assume x <= y & z <= t;
then x+z <= y+z & y+z <= y+t by Th32;
hence thesis by SURREALO:4;
end;
theorem Th44:
x <= y & z < t implies x+z < y+t
proof
assume A1: x <= y & z < t;
assume not x+z < y+t;
then A2: z >= y+t - x by Th42;
A3: x-x ==0_No by Th39;
x+-x <= y +-x by A1,Th32;
then 0_No <= y +-x by A3,SURREALO:4;
then A4: t+0_No <= t+(y +-x) by Th32;
y+t +- x = t+(y +- x) by Th37;
hence thesis by A1,A2,A4,SURREALO:4;
end;
theorem
x < y iff 0_No < y - x
proof
x=0_No+x;
hence thesis by Th42;
end;
theorem
x < y iff x - y < 0_No
proof
y=0_No+y;
hence thesis by Th41;
end;
theorem
x - y == 0_No implies x == y
proof
A1: -y+y=y-y ==0_No by Th39;
assume x - y == 0_No;
then y = 0_No+ y == x + - y + y = x + (- y + y) by Th37,Th43;
then y == x +(-y+y) == x+0_No =x by A1,Th43;
hence thesis by SURREALO:4;
end;
definition
let x be object;
assume A1:x is surreal;
func -' x -> Surreal means :Def10:
for x1 be Surreal st x1=x holds it = - x1;
existence
proof
reconsider xx=x as Surreal by A1;
take -xx;
thus thesis;
end;
uniqueness
proof
reconsider xx=x as Surreal by A1;
let m1,m2 be Surreal such that
A2: for x1 be Surreal st x1=x holds m1 = - x1 and
A3: for x1 be Surreal st x1=x holds m2 = - x1;
m1 = -xx by A2;
hence thesis by A3;
end;
end;
registration
let a be Surreal, x be object;
identify -a with -' x when a = x;
compatibility by Def10;
end;
definition
let x,y be object;
assume A1:x is surreal & y is surreal;
func x +' y -> Surreal means :Def11:
for x1,y1 be Surreal st x1=x & y1=y holds it = x1+y1;
existence
proof
reconsider xx=x,yy=y as Surreal by A1;
take xx+yy;
thus thesis;
end;
uniqueness
proof
reconsider xx=x,yy=y as Surreal by A1;
let a1,a2 be Surreal such that
A2: for x1,y1 be Surreal st x1=x & y1=y holds a1 = x1+y1 and
A3: for x1,y1 be Surreal st x1=x & y1=y holds a2 = x1+y1;
a1 = xx+yy by A2;
hence thesis by A3;
end;
end;
registration
let a,b be Surreal, x,y be object;
identify a+b with x+'y when a = x, b = y;
compatibility by Def11;
end;
begin :: The Product of Superreal Numbers
definition
let A be Ordinal;
func No_mult_op A -> ManySortedSet of Triangle A means :Def12:
ex S be c=-monotone Function-yielding Sequence st
dom S = succ A & it = S.A &
for B be Ordinal st B in succ A
ex SB be ManySortedSet of Triangle B st S.B = SB &
for x be object st x in Triangle B holds
SB.x = [ {(union rng (S|B).[xL,R_x]) +'
(union rng (S|B).[L_x,yL]) +' -'
(union rng (S|B).[xL,yL])
where xL is Element of L_L_x, yL is Element of L_R_x:
xL in L_L_x & yL in L_R_x}
\/
{(union rng (S|B).[xR,R_x]) +'
(union rng (S|B).[L_x,yR]) +' -'
(union rng (S|B).[xR,yR])
where xR is Element of R_L_x, yR is Element of R_R_x:
xR in R_L_x & yR in R_R_x}
,
{(union rng (S|B).[xL,R_x]) +'
(union rng (S|B).[L_x,yR]) +' -'
(union rng (S|B).[xL,yR])
where xL is Element of L_L_x, yR is Element of R_R_x:
xL in L_L_x & yR in R_R_x}
\/
{(union rng (S|B).[xR,R_x]) +'
(union rng (S|B).[L_x,yL]) +' -'
(union rng (S|B).[xR,yL])
where xR is Element of R_L_x, yL is Element of L_R_x:
xR in R_L_x & yL in L_R_x}
];
existence
proof
deffunc D(Ordinal) = Triangle $1;
deffunc H(object,c=-monotone Function-yielding Sequence) =
[ {(union rng $2.[xL,R_$1]) +'
(union rng $2.[L_$1,yL]) +' -' (union rng $2.[xL,yL])
where xL is Element of L_L_$1,yL is Element of L_R_$1:
xL in L_L_$1 & yL in L_R_$1} \/
{(union rng $2.[xR,R_$1]) +'
(union rng $2.[L_$1,yR]) +' -' (union rng $2.[xR,yR])
where xR is Element of R_L_$1,yR is Element of R_R_$1:
xR in R_L_$1 & yR in R_R_$1},
{(union rng $2.[xL,R_$1]) +'
(union rng $2.[L_$1,yR]) +' -' (union rng $2.[xL,yR])
where xL is Element of L_L_$1,yR is Element of R_R_$1:
xL in L_L_$1 & yR in R_R_$1}
\/
{(union rng $2.[xR,R_$1]) +'
(union rng $2.[L_$1,yL]) +' -' (union rng $2.[xR,yL])
where xR is Element of R_L_$1,yL is Element of L_R_$1:
xR in R_L_$1 & yL in L_R_$1}];
A1: for A,B be Ordinal st A c= B holds D(A) c= D(B) by Th25;
A2: for S be c=-monotone Function-yielding Sequence st
for A be Ordinal st A in dom S holds dom (S.A) = D(A)
for A be Ordinal,x be object st x in dom (S.A) holds H(x,S|A) = H(x,S)
proof
let S be c=-monotone Function-yielding Sequence such that
A3:for A be Ordinal st A in dom S holds dom (S.A) = D(A);
let A be Ordinal,a be object such that A4: a in dom (S.A);
S.A <>{} by A4;
then A5:A in dom S by FUNCT_1:def 2;
then A6: dom (S.A) = D(A) by A3;
then consider x,y be object such that
A7:x in Day A & y in Day A & a=[x,y] by A4,ZFMISC_1:def 2;
reconsider x,y as Surreal by A7;
A8: born x (+) born y c= A by A7,A6,A4,Def5;
A9 : for X,Y be surreal-membered set st X c= L_x\/R_x & Y c= L_y\/R_y
holds
{(union rng (S|A).[xL,y]) +'(union rng (S|A).[x,yL]) +'
-' (union rng (S|A).[xL,yL]) where xL is Element of X,yL is Element of Y:
xL in X & yL in Y}
=
{(union rng S.[xL,y]) +'(union rng S.[x,yL]) +' -' (union rng S.[xL,yL])
where xL is Element of X,yL is Element of Y: xL in X & yL in Y}
proof
let X,Y be surreal-membered set such that
A10: X c= L_x\/R_x & Y c= L_y\/R_y;
thus {(union rng (S|A).[xL,y]) +'
(union rng (S|A).[x,yL]) +' -' (union rng (S|A).[xL,yL])
where xL is Element of X,yL is Element of Y: xL in X & yL in Y}
c=
{(union rng S.[xL,y]) +' (union rng S.[x,yL])
+' -' (union rng S.[xL,yL])
where xL is Element of X,yL is Element of Y: xL in X & yL in Y}
proof
let z be object;
assume z in {(union rng (S|A).[xL,y]) +'
(union rng (S|A).[x,yL]) +' -' (union rng (S|A).[xL,yL])
where xL is Element of X,yL is Element of Y: xL in X & yL in Y};
then consider xL be Element of X,yL be Element of Y such that
A11: z = (union rng (S|A).[xL,y]) +' (union rng (S|A).[x,yL]) +'
-' (union rng (S|A).[xL,yL]) & xL in X & yL in Y;
reconsider xxL=xL,yyL=yL as Surreal by SURREAL0:def 16,A11;
set Lx=born xxL (+) born y;
A12:Lx in born x (+) born y by ORDINAL7:94,SURREALO:1,A10,A11;
then A13:dom (S.Lx) = D(Lx) by A8,A3,A5,ORDINAL1:10;
[xL,y] in D(Lx) by Def5;
then A14:union rng (S|A).[xL,y] = (union rng S).[xL,y]
by Th6,A13,A12,A8;
set Ly=born x (+) born yyL;
A15:Ly in born x (+) born y by A10,A11,ORDINAL7:94,SURREALO:1;
then A16:dom (S.Ly) = D(Ly) by A8,A3,A5,ORDINAL1:10;
[x,yL] in D(Ly) by Def5;
then A17: union rng (S|A).[x,yL] = (union rng S).[x,yL]
by Th6,A16,A15,A8;
set Lxy=born xxL (+) born yyL;
A18: Lxy in Ly by A10,A11,ORDINAL7:94,SURREALO:1;
then Lxy in born x (+) born y by A15,ORDINAL1:10;
then A19:dom (S.Lxy) = D(Lxy) by A8,A3,A5,ORDINAL1:10;
[xL,yL] in D(Lxy) by Def5;
then union rng (S|A).[xL,yL] = (union rng S).[xL,yL]
by Th6,A19,A18,A8,A15,ORDINAL1:10;
hence thesis by A14,A17,A11;
end;
let z be object;
assume z in {(union rng S.[xL,y]) +'
(union rng S.[x,yL]) +' -' (union rng S.[xL,yL])
where xL is Element of X,yL is Element of Y:xL in X & yL in Y};
then consider xL be Element of X,yL be Element of Y such that
A20: z = (union rng (S).[xL,y]) +'
(union rng (S).[x,yL]) +' -' (union rng (S).[xL,yL]) &
xL in X & yL in Y;
reconsider xxL=xL,yyL=yL as Surreal by SURREAL0:def 16,A20;
set Lx=born xxL (+) born y;
A21:Lx in born x (+) born y by A10,A20,ORDINAL7:94,SURREALO:1;
then A22:dom (S.Lx) = D(Lx) by A8,A3,A5,ORDINAL1:10;
[xL,y] in D(Lx) by Def5;
then A23:union rng (S|A).[xL,y] = (union rng S).[xL,y]
by Th6,A22,A21,A8;
set Ly=born x (+) born yyL;
A24:Ly in born x (+) born y by A10,A20,ORDINAL7:94,SURREALO:1;
then A25:dom (S.Ly) = D(Ly) by A8,A3,A5,ORDINAL1:10;
[x,yL] in D(Ly) by Def5;
then A26: union rng (S|A).[x,yL] = (union rng S).[x,yL]
by Th6,A25,A24,A8;
set Lxy=born xxL (+) born yyL;
A27: Lxy in Ly by A10,A20,ORDINAL7:94,SURREALO:1;
then Lxy in born x (+) born y by A24,ORDINAL1:10;
then A28:dom (S.Lxy) = D(Lxy) by A8,A3,A5,ORDINAL1:10;
[xL,yL] in D(Lxy) by Def5;
then union rng (S|A).[xL,yL] = (union rng S).[xL,yL]
by Th6,A28,A27,A8,A24,ORDINAL1:10;
hence thesis by A23,A26,A20;
end;
A29:L_x c= L_x \/ R_x & R_x c= L_x \/ R_x &
L_y c= L_y\/R_y & R_y c= L_y\/R_y by XBOOLE_1:7;
then A30: {(union rng (S|A).[xL,y]) +' (union rng (S|A).[x,yL]) +' -'
(union rng (S|A).[xL,yL])
where xL is Element of L_x,yL is Element of L_y: xL in L_x & yL in L_y}
=
{(union rng S.[xL,y]) +'
(union rng S.[x,yL]) +' -' (union rng S.[xL,yL])
where xL is Element of L_x,yL is Element of L_y:
xL in L_x & yL in L_y} by A9;
A31: {(union rng (S|A).[xR,y]) +' (union rng (S|A).[x,yR]) +'
-' (union rng (S|A).[xR,yR])
where xR is Element of R_x,yR is Element of R_y: xR in R_x & yR in R_y}
=
{(union rng (S).[xR,y]) +'
(union rng (S).[x,yR]) +' -' (union rng (S).[xR,yR])
where xR is Element of R_x,yR is Element of R_y:
xR in R_x & yR in R_y} by A29,A9;
{(union rng (S|A).[xL,y]) +'
(union rng (S|A).[x,yR]) +' -' (union rng (S|A).[xL,yR])
where xL is Element of L_x,yR is Element of R_y: xL in L_x & yR in R_y}
= {(union rng (S).[xL,y]) +'
(union rng (S).[x,yR]) +' -' (union rng S.[xL,yR])
where xL is Element of L_x,yR is Element of R_y:
xL in L_x & yR in R_y} by A29,A9;
hence thesis by A7,A30,A31,A29,A9;
end;
consider S be c=-monotone Function-yielding Sequence such that
A32: dom S = succ A and
A33: for A1 be Ordinal st A1 in succ A
ex SA1 be ManySortedSet of D(A1) st S.A1 = SA1 &
for x be object st x in D(A1) holds SA1.x = H(x,S|A1)
from MonoFvSExists(A2,A1);
consider SA be ManySortedSet of D(A) such that
A34:S.A = SA and
for x be object st x in D(A) holds SA.x = H(x,S|A) by A33,ORDINAL1:6;
take SA,S;
thus thesis by A32,A34,A33;
end;
uniqueness
proof
deffunc D(Ordinal) = Triangle $1;
deffunc H(object,c=-monotone Function-yielding Sequence) =
[ {(union rng $2.[xL,R_$1]) +'
(union rng $2.[L_$1,yL]) +' -' (union rng $2.[xL,yL])
where xL is Element of L_L_$1,yL is Element of L_R_$1:
xL in L_L_$1 & yL in L_R_$1} \/
{(union rng $2.[xR,R_$1]) +'
(union rng $2.[L_$1,yR]) +' -' (union rng $2.[xR,yR])
where xR is Element of R_L_$1,yR is Element of R_R_$1:
xR in R_L_$1 & yR in R_R_$1},
{(union rng $2.[xL,R_$1]) +'
(union rng $2.[L_$1,yR]) +' -' (union rng $2.[xL,yR])
where xL is Element of L_L_$1,yR is Element of R_R_$1:
xL in L_L_$1 & yR in R_R_$1}
\/
{(union rng $2.[xR,R_$1]) +'
(union rng $2.[L_$1,yL]) +' -' (union rng $2.[xR,yL])
where xR is Element of R_L_$1,yL is Element of L_R_$1:
xR in R_L_$1 & yL in L_R_$1}];
let it1,it2 be ManySortedSet of Triangle A;
given S1 be c=-monotone Function-yielding Sequence such that
A35:dom S1 = succ A & it1 = S1.A and
A36:for B be Ordinal st B in succ A
ex SB be ManySortedSet of D(B) st S1.B = SB &
for x be object st x in D(B) holds SB.x = H(x,S1|B);
given S2 be c=-monotone Function-yielding Sequence such that
A37:dom S2 = succ A & it2 = S2.A and
A38:for B be Ordinal st B in succ A
ex SB be ManySortedSet of D(B) st S2.B = SB &
for x be object st x in D(B) holds SB.x = H(x,S2|B);
A39:succ A c= dom S1 & succ A c= dom S2 by A35,A37;
S1|succ A = S2|succ A from MonoFvSUniq(A39,A36,A38);
then S1|succ A = S2 by A37;
hence thesis by A35,A37;
end;
end;
definition
let x,y be Surreal;
func x * y -> set equals
(No_mult_op (born x (+) born y)).[x,y];
coherence;
end;
theorem Th48:
for S be c=-monotone Function-yielding Sequence st
for B be Ordinal st B in dom S
ex SB be ManySortedSet of Triangle B st S.B = SB &
for x be object st x in Triangle B holds
SB.x = [ {(union rng (S|B).[xL,R_x]) +'
(union rng (S|B).[L_x,yL]) +' -'
(union rng (S|B).[xL,yL])
where xL is Element of L_L_x, yL is Element of L_R_x:
xL in L_L_x & yL in L_R_x}
\/
{(union rng (S|B).[xR,R_x]) +'
(union rng (S|B).[L_x,yR]) +' -'
(union rng (S|B).[xR,yR])
where xR is Element of R_L_x, yR is Element of R_R_x:
xR in R_L_x & yR in R_R_x}
,
{(union rng (S|B).[xL,R_x]) +'
(union rng (S|B).[L_x,yR]) +' -'
(union rng (S|B).[xL,yR])
where xL is Element of L_L_x, yR is Element of R_R_x:
xL in L_L_x & yR in R_R_x}
\/
{(union rng (S|B).[xR,R_x]) +'
(union rng (S|B).[L_x,yL]) +' -'
(union rng (S|B).[xR,yL])
where xR is Element of R_L_x, yL is Element of L_R_x:
xR in R_L_x & yL in L_R_x}
]
holds
for A be Ordinal st A in dom S holds No_mult_op A = S.A
proof
deffunc D(Ordinal) = Triangle $1;
deffunc H(object,c=-monotone Function-yielding Sequence) =
[ {(union rng $2.[xL,R_$1]) +'
(union rng $2.[L_$1,yL]) +' -' (union rng $2.[xL,yL])
where xL is Element of L_L_$1,yL is Element of L_R_$1:
xL in L_L_$1 & yL in L_R_$1} \/
{(union rng $2.[xR,R_$1]) +'
(union rng $2.[L_$1,yR]) +' -' (union rng $2.[xR,yR])
where xR is Element of R_L_$1,yR is Element of R_R_$1:
xR in R_L_$1 & yR in R_R_$1},
{(union rng $2.[xL,R_$1]) +'
(union rng $2.[L_$1,yR]) +' -' (union rng $2.[xL,yR])
where xL is Element of L_L_$1,yR is Element of R_R_$1:
xL in L_L_$1 & yR in R_R_$1}
\/
{(union rng $2.[xR,R_$1]) +'
(union rng $2.[L_$1,yL]) +' -' (union rng $2.[xR,yL])
where xR is Element of R_L_$1,yL is Element of L_R_$1:
xR in R_L_$1 & yL in L_R_$1}];
let S be c=-monotone Function-yielding Sequence such that
A1: for B be Ordinal st B in dom S
ex SB be ManySortedSet of D(B) st S.B = SB &
for x be object st x in D(B) holds SB.x = H(x,S|B);
let A be Ordinal such that A2: A in dom S;
consider S1 be c=-monotone Function-yielding Sequence such that
A3:dom S1 = succ A & No_mult_op A = S1.A and
A4:for B be Ordinal st B in succ A
ex SB be ManySortedSet of D(B) st S1.B = SB &
for x be object st x in D(B) holds SB.x = H(x,S1|B) by Def12;
A5:succ A c= dom S by A2,ORDINAL1:21;
A6: for B be Ordinal st B in succ A
ex SB be ManySortedSet of D(B) st S.B = SB &
for x be object st x in D(B) holds SB.x = H(x,S|B) by A1,A5;
A7: succ A c= dom S & succ A c= dom S1 by A2,ORDINAL1:21,A3;
A8: S|succ A = S1|succ A from MonoFvSUniq(A7,A6,A4);
A in succ A by ORDINAL1:8;
hence thesis by A3,A8,FUNCT_1:49;
end;
definition
let x,y be Surreal;
let X,Y be set;
func comp(X,x,y,Y) -> set means :Def14:
o in it iff ex x1,y1 be Surreal st
o = (x1*y) +' (x*y1) +' -' (x1*y1) & x1 in X & y1 in Y;
existence
proof
defpred P[object] means $1 is surreal;
consider X1 be set such that
A1:for o holds o in X1 iff o in X & P[o] from XBOOLE_0:sch 1;
consider Y1 be set such that
A2:for o holds o in Y1 iff o in Y & P[o] from XBOOLE_0:sch 1;
defpred Py[object,object] means
for a be Surreal st a =$1 holds $2 = a*y;
defpred Px[object,object] means
for a be Surreal st a =$1 holds $2 = x*a;
A3:for o be object st o in X1 ex u being object st Py[o,u]
proof
let o be object such that A4:o in X1;
reconsider a=o as Surreal by A4,A1;
take a*y;
thus thesis;
end;
consider fy be Function such that
A5: dom fy = X1 & for o be object st o in X1 holds Py[o,fy.o]
from CLASSES1:sch 1(A3);
A6:for o be object st o in Y1 ex u being object st Px[o,u]
proof
let o be object such that A7:o in Y1;
reconsider a=o as Surreal by A7,A2;
take x*a;
thus thesis;
end;
consider fx be Function such that
A8: dom fx = Y1 & for o be object st o in Y1 holds Px[o,fx.o]
from CLASSES1:sch 1(A6);
defpred Pxy[object,object] means
for a,b be Surreal st $1=[a,b] holds $2 = a*b;
A9:for o be object st o in [:X1,Y1:] ex u being object st Pxy[o,u]
proof
let o be object such that A10:o in [:X1,Y1:];
consider a,b be object such that
A11:a in X1 & b in Y1 & o = [a,b] by A10,ZFMISC_1:def 2;
reconsider a,b as Surreal by A11,A1,A2;
take a*b;
let a1,b1 be Surreal;assume o=[a1,b1];
then a1=a & b1=b by A11,XTUPLE_0:1;
hence thesis;
end;
consider fxy be Function such that
A12: dom fxy = [:X1,Y1:] &
for o be object st o in [:X1,Y1:] holds Pxy[o,fxy.o]
from CLASSES1:sch 1(A9);
set IT={fy.x1 +' fx.y1 +' -' fxy.[x1,y1]
where x1 is Element of X,y1 is Element of Y:
x1 in X & y1 in Y & x1 is surreal & y1 is surreal};
take IT;
let o;
thus o in IT implies ex x1,y1 be Surreal st
o = (x1*y) +' (x*y1) +' -' (x1*y1) & x1 in X & y1 in Y
proof
assume o in IT;
then consider x1 be Element of X,y1 be Element of Y such that
A13: o = fy.x1 +' fx.y1 +' -' fxy.[x1,y1] & x1 in X & y1 in Y &
x1 is surreal & y1 is surreal;
reconsider x1,y1 as Surreal by A13;
x1 in X1 & y1 in Y1 by A13,A1,A2;
then [x1,y1] in [:X1,Y1:] by ZFMISC_1:def 2;
then A14:fxy.[x1,y1] = x1*y1 by A12;
A15:fy.x1 = x1*y by A13,A1,A5;
fx.y1 = x*y1 by A13,A2,A8;
hence thesis by A13,A14,A15;
end;
given x1,y1 be Surreal such that
A16: o = x1*y +' (x*y1) +' -' x1*y1 & x1 in X & y1 in Y;
x1 in X1 & y1 in Y1 by A16,A1,A2;
then [x1,y1] in [:X1,Y1:] by ZFMISC_1:def 2;
then A17:fxy.[x1,y1] = x1*y1 by A12;
A18:fy.x1 = x1*y by A16,A1,A5;
fx.y1 = x*y1 by A16,A2,A8;
hence thesis by A16,A17,A18;
end;
uniqueness
proof
let C1,C2 be set such that
A19:o in C1 iff ex x1,y1 be Surreal st
o = x1*y +' (x*y1) +' -' x1*y1 & x1 in X & y1 in Y and
A20:o in C2 iff ex x1,y1 be Surreal st
o = x1*y +' (x*y1) +' -' x1*y1 & x1 in X & y1 in Y;
now let o;
o in C1 iff ex x1,y1 be Surreal st
o = x1*y +' (x*y1) +' -' x1*y1 & x1 in X & y1 in Y by A19;
hence o in C1 iff o in C2 by A20;
end;
hence thesis by TARSKI:2;
end;
end;
theorem Th49:
for X be set holds comp(X,x,y,{}) = {}
proof
let X be set;
assume comp(X,x,y,{}) <> {};
then consider o such that
A1: o in comp(X,x,y,{}) by XBOOLE_0:def 1;
consider x1,y1 be Surreal such that
A2: o = (x1*y) +' (x*y1) +' -' (x1*y1) & x1 in X & y1 in {} by A1,Def14;
thus thesis by A2;
end;
theorem Th50:
for x,y be Surreal holds
x * y = [comp(L_x,x,y,L_y) \/ comp(R_x,x,y,R_y),
comp(L_x,x,y,R_y) \/ comp(R_x,x,y,L_y)]
proof
let x,y be Surreal;
set Bx=born x,By=born y,A=Bx (+)By;
deffunc UU(object,object,c=-monotone Function-yielding Sequence) =
[( (union rng ($3).:[:L_$1,{$2}:]) ++
(union rng ($3).:[:{$1},L_$2:]) ++ --
(union rng ($3).:[:L_$1,L_$2:]) ) \/
( (union rng ($3).:[:R_$1,{$2}:]) ++
(union rng ($3).:[:{$1},R_$2:]) ++ --
(union rng ($3).:[:R_$1,R_$2:])),
( (union rng ($3).:[:L_$1,{$2}:]) ++
(union rng ($3).:[:{$1},R_$2:]) ++ --
(union rng ($3).:[:L_$1,R_$2:]) ) \/
( (union rng ($3).:[:R_$1,{$2}:]) ++
(union rng ($3).:[:{$1},L_$2:]) ++ --
(union rng ($3).:[:R_$1`2,L_$2:]))];
deffunc H(object,c=-monotone Function-yielding Sequence) =
[ {(union rng $2.[xL,R_$1]) +'
(union rng $2.[L_$1,yL]) +' -' (union rng $2.[xL,yL])
where xL is Element of L_L_$1,yL is Element of L_R_$1:
xL in L_L_$1 & yL in L_R_$1} \/
{(union rng $2.[xR,R_$1]) +'
(union rng $2.[L_$1,yR]) +' -' (union rng $2.[xR,yR])
where xR is Element of R_L_$1,yR is Element of R_R_$1:
xR in R_L_$1 & yR in R_R_$1},
{(union rng $2.[xL,R_$1]) +'
(union rng $2.[L_$1,yR]) +' -' (union rng $2.[xL,yR])
where xL is Element of L_L_$1,yR is Element of R_R_$1:
xL in L_L_$1 & yR in R_R_$1}
\/
{(union rng $2.[xR,R_$1]) +'
(union rng $2.[L_$1,yL]) +' -' (union rng $2.[xR,yL])
where xR is Element of R_L_$1,yL is Element of L_R_$1:
xR in R_L_$1 & yL in L_R_$1}];
consider S be c=-monotone Function-yielding Sequence such that
A1: dom S = succ A & No_mult_op A = S.A and
A2: for B be Ordinal st B in succ A
ex SB be ManySortedSet of Triangle B st S.B = SB &
for x be object st x in Triangle B holds SB.x = H(x,S|B) by Def12;
consider SA be ManySortedSet of Triangle A such that
A3:S.A = SA and
A4:for x be object st x in Triangle A holds SA.x = H(x,S|A)
by A2,ORDINAL1:6;
set U =union rng (S|A);
A5:[x,y] in Triangle A by Def5;
A6: x*y = H([x,y],S|A) by A1,A3,A5,A4;
A7:for X,Y be surreal-membered set st X c= L_x\/ R_x & Y c= L_y\/ R_y
holds {(U.[xL,y]) +' (U.[x,yL]) +' -' (U.[xL,yL])
where xL is Element of X,yL is Element of Y:xL in X & yL in Y}
= comp(X,x,y,Y)
proof
let X,Y be surreal-membered set such that
A8: X c= L_x\/ R_x & Y c= L_y \/R_y;
thus {(U.[xL,y]) +' (U.[x,yL]) +' -' (U.[xL,yL])
where xL is Element of X,yL is Element of Y:xL in X & yL in Y}
c= comp(X,x,y,Y)
proof
let a be object;
assume a in {(U.[xL,y]) +' (U.[x,yL]) +' -' (U.[xL,yL])
where xL is Element of X,yL is Element of Y:xL in X & yL in Y};
then consider xL be Element of X,yL be Element of Y such that
A9:a= (U.[xL,y])+' (U.[x,yL])+' -' (U.[xL,yL]) & xL in X & yL in Y;
reconsider x1=xL,y1=yL as Surreal by SURREAL0:def 16,A9;
set X1 = born x1,C1 = X1 (+) born y;
A10: C1 in A by A9,A8,SURREALO:1,ORDINAL7:94;
then A11: C1 in succ A by ORDINAL1:8;
then consider SC1 be ManySortedSet of Triangle C1 such that
A12:S.C1 = SC1 and
for x be object st x in Triangle C1 holds
SC1.x = H(x,S|C1) by A2;
A13:dom SC1 = Triangle C1 by PARTFUN1:def 2;
A14: [x1,y] in Triangle C1 by Def5;
A15: x1 * y = SC1.[x1,y] by A11,A1,A2,A12,Th48;
A16: SC1. [x1,y] = (union rng S).[x1,y] by A11,A13,A14,Th2,A1,A12;
A17: U. [x1,y] = x1*y by A16,A15,A12,Th5,A10,A14,A13;
set Y1 = born y1,C2 = born x (+) Y1;
A18: C2 in A by A9,A8,SURREALO:1,ORDINAL7:94;
then A19: C2 in succ A by ORDINAL1:8;
then consider SC2 be ManySortedSet of Triangle C2 such that
A20:S.C2 = SC2 and
for x be object st x in Triangle C2 holds
SC2.x = H(x,S|C2) by A2;
A21:dom SC2 = Triangle C2 by PARTFUN1:def 2;
A22: [x,y1] in Triangle C2 by Def5;
A23: x * y1 = SC2.[x,y1] by A19,A1,A2,A20,Th48;
A24: SC2. [x,y1] = (union rng S).[x,y1] by A19,A21,A22,Th2,A1,A20;
A25: U. [x,y1] = x*y1 by A24,A23,A20,Th5,A18,A22,A21;
set C3 = X1 (+) Y1;
C3 in C2 in A by A9,A8,SURREALO:1,ORDINAL7:94;
then A26: C3 in A by ORDINAL1:10;
then A27: C3 in succ A by ORDINAL1:8;
then consider SC3 be ManySortedSet of Triangle C3 such that
A28:S.C3 = SC3 and
for x be object st x in Triangle C3 holds
SC3.x = H(x,S|C3) by A2;
A29:dom SC3 = Triangle C3 by PARTFUN1:def 2;
A30: [x1,y1] in Triangle C3 by Def5;
A31:x1 * y1 = SC3.[x1,y1] by A27,A1,A2,A28,Th48;
A32:SC3. [x1,y1] = (union rng S).[x1,y1] by A27,A29,A30,Th2,A1,A28;
a= (U.[xL,y]) +' (U.[x,yL]) +' -' (x1*y1)
by A9,A32,A31,A28,Th5,A26,A30,A29;
hence thesis by Def14,A9,A17,A25;
end;
let a be object;
assume a in comp(X,x,y,Y);
then consider x1,y1 be Surreal such that
A33: a = (x1*y) +' (x*y1) +' -' (x1*y1) & x1 in X & y1 in Y by Def14;
set X1 = born x1,C1 = X1 (+) born y;
A34:C1 in A by A33,A8,SURREALO:1,ORDINAL7:94;
then A35: C1 in succ A by ORDINAL1:8;
then consider SC1 be ManySortedSet of Triangle C1 such that
A36:S.C1 = SC1 and
for x be object st x in Triangle C1 holds
SC1.x = H(x,S|C1) by A2;
A37:dom SC1 = Triangle C1 by PARTFUN1:def 2;
A38: [x1,y] in Triangle C1 by Def5;
A39: x1 * y = SC1.[x1,y] by A35,A1,A2,A36,Th48;
A40: SC1. [x1,y] = (union rng S).[x1,y] by A35,A37,A38,Th2,A1,A36;
A41: U. [x1,y] = x1*y by A40,A39,A36,Th5,A34,A38,A37;
set Y1 = born y1,C2 = born x (+) Y1;
A42: C2 in A by A33,A8,SURREALO:1,ORDINAL7:94;
then A43: C2 in succ A by ORDINAL1:8;
then consider SC2 be ManySortedSet of Triangle C2 such that
A44:S.C2 = SC2 and
for x be object st x in Triangle C2 holds
SC2.x = H(x,S|C2) by A2;
A45:dom SC2 = Triangle C2 by PARTFUN1:def 2;
A46: [x,y1] in Triangle C2 by Def5;
A47: x * y1 = SC2.[x,y1] by A43,A1,A2,A44,Th48;
A48: SC2. [x,y1] = (union rng S).[x,y1] by A43,A45,A46,Th2,A1,A44;
A49: U. [x,y1] = x*y1 by A48,A47,A44,Th5,A42,A46,A45;
set C3 = X1 (+) Y1;
C3 in C2 in A by A33,A8,SURREALO:1,ORDINAL7:94;
then A50: C3 in A by ORDINAL1:10;
then A51: C3 in succ A by ORDINAL1:8;
then consider SC3 be ManySortedSet of Triangle C3 such that
A52:S.C3 = SC3 and
for x be object st x in Triangle C3 holds
SC3.x = H(x,S|C3)by A2;
A53:dom SC3 = Triangle C3 by PARTFUN1:def 2;
A54: [x1,y1] in Triangle C3 by Def5;
A55: x1 * y1 = SC3.[x1,y1] by A51,A1,A2,A52,Th48;
A56: SC3. [x1,y1] = (union rng S).[x1,y1] by A51,A53,A54,Th2,A1,A52;
a= (x1*y) +' (x*y1) +' -' (U. [x1,y1])
by A33,A56,A55,A52,Th5,A50,A54,A53;
hence thesis by A33,A49,A41;
end;
A57:L_x c= L_x \/ R_x & R_x c= L_x \/ R_x &
L_y c= L_y\/R_y & R_y c= L_y\/R_y by XBOOLE_1:7;
A58:{(U.[xL,y]) +'(U.[x,yL]) +' -' (U.[xL,yL])
where xL is Element of L_x,yL is Element of L_y:xL in L_x & yL in L_y}
=comp(L_x,x,y,L_y) by A57,A7;
A59:{(U.[xL,y]) +'(U.[x,yR]) +' -' (U.[xL,yR])
where xL is Element of L_x,yR is Element of R_y:xL in L_x & yR in R_y}
=comp(L_x,x,y,R_y) by A57,A7;
{(U.[xR,y]) +'(U.[x,yR]) +' -' (U.[xR,yR])
where xR is Element of R_x,yR is Element of R_y:xR in R_x & yR in R_y}
=comp(R_x,x,y,R_y) by A57,A7;
hence thesis by A6,A58,A59,A57,A7;
end;
theorem Th51:
(for x,y holds x*y is Surreal) &
(for x,y holds x*y = y*x)&
(for x1,x2,y,x1y,x2y be Surreal st x1 == x2 & x1y = x1*y & x2y = x2*y
holds x1y == x2y) &
(for x1,x2,y1,y2,x1y2,x2y1,x1y1,x2y2 be Surreal st
x1y1=x1*y1 & x1y2=x1*y2 & x2y1=x2*y1 & x2y2=x2*y2 & x1 < x2 & y1 < y2
holds x1y2+x2y1 < x1y1+x2y2)
proof
defpred P0[Ordinal,Surreal,Surreal] means
born $2 (+) born $3 c= $1 implies $2*$3 = $3 * $2;
defpred P1[Ordinal,Surreal,Surreal] means
born $2 (+) born $3 c= $1 implies $2*$3 is Surreal;
defpred P2[Ordinal,Surreal,Surreal,Surreal] means
for x1y,x2y be Surreal st
born $2 (+) born $4 c= $1 & born $3 (+) born $4 c= $1 &
$2 == $3 & x1y = $2*$4 & x2y = $3*$4
holds x1y == x2y;
defpred P3[Ordinal,Surreal,Surreal,Surreal,Surreal] means
for x1y2,x2y1,x1y1,x2y2 be Surreal st
born $2 (+) born $4 c= $1 & born $3 (+) born $4 c= $1 &
born $2 (+) born $5 c= $1 & born $3 (+) born $5 c= $1 &
x1y1=$2*$4 & x1y2=$2*$5 & x2y1=$3*$4 & x2y2=$3*$5 & $2 < $3 & $4 < $5
holds
x1y2+x2y1 < x1y1+x2y2;
defpred P4[Ordinal,Surreal,Surreal,Surreal] means
$3 < $4 implies (for x st x in L_$2 holds P3[$1,x,$2,$3,$4])&
(for x st x in R_$2 holds P3[$1,$2,x,$3,$4]);
defpred Q0[Ordinal] means for x,y holds P0[$1,x,y];
defpred Q1[Ordinal] means for x,y holds P1[$1,x,y];
defpred Q2[Ordinal] means for x1,x2,y be Surreal holds P2[$1,x1,x2,y];
defpred Q3[Ordinal] means for x1,x2,y1,y2 be Surreal
holds P3[$1,x1,x2,y1,y2];
defpred Q[Ordinal] means Q0[$1] & Q1[$1] & Q2[$1] & Q3[$1];
A1: for D be Ordinal st for C be Ordinal st C in D holds Q[C] holds Q[D]
proof
let D be Ordinal such that
A2: for C be Ordinal st C in D holds Q[C];
thus A3: for x,y holds P0[D,x,y]
proof
A4: for x,y st born x (+) born y c= D
for X,Y be surreal-membered set st (X c= L_x\/R_x & Y c= L_y\/R_y)
holds comp(X,x,y,Y) c= comp(Y,y,x,X)
proof
let x,y be Surreal such that A5: born x (+) born y c= D;
let X,Y be surreal-membered set such that
A6:X c= L_x\/R_x & Y c= L_y\/R_y;
let a be object;
assume a in comp(X,x,y,Y);
then consider x1,y1 be Surreal such that
A7: a = (x1*y) +' (x*y1) +' -' (x1*y1) & x1 in X & y1 in Y by Def14;
A8: born x1 (+) born y1 in born x1 (+) born y &
born x (+) born y1 in born x (+) born y &
born x1 (+) born y in born x (+) born y
by SURREALO:1,A6,A7,ORDINAL7:94;
then born x1 (+) born y1 in born x (+) born y by ORDINAL1:10;
then A9:x1*y1 = y1*x1 & x*y1 = y1*x & x1*y = y*x1 by A8,A2,A5;
reconsider yx1=y*x1,y1x=y1*x as Surreal by A8,A2,A5;
(y*x1) +' (y1*x) =y1x +yx1 by Def11
.=(y1*x) +' (y*x1);
hence thesis by A7,Def14,A9;
end;
A10:for x,y st born x (+) born y c= D
for X,Y be surreal-membered set st (X c= L_x\/R_x & Y c= L_y\/R_y)
holds comp(X,x,y,Y) = comp(Y,y,x,X)
proof
let x,y such that A11: born x (+) born y c= D;
let X,Y be surreal-membered set such that
A12: X c= L_x\/R_x & Y c= L_y\/R_y;
comp(X,x,y,Y) c= comp(Y,y,x,X) c= comp(X,x,y,Y)by A11,A12, A4;
hence thesis by XBOOLE_0:def 10;
end;
let x,y be Surreal such that A13: born x (+) born y c= D;
L_x c= L_x \/ R_x & R_x c= L_x \/ R_x &
L_y c= L_y\/R_y & R_y c= L_y\/R_y by XBOOLE_1:7;
then A14: comp(L_x,x,y,L_y) = comp(L_y,y,x,L_x) &
comp(R_x,x,y,R_y) = comp(R_y,y,x,R_x) &
comp(L_x,x,y,R_y) = comp(R_y,y,x,L_x) &
comp(R_x,x,y,L_y) = comp(L_y,y,x,R_x) by A10,A13;
x*y = [comp(L_x,x,y,L_y) \/ comp(R_x,x,y,R_y),
comp(L_x,x,y,R_y) \/ comp(R_x,x,y,L_y)] by Th50
.=y*x by A14,Th50;
hence thesis;
end;
thus for x,y holds P1[D,x,y]
proof
let x,y;
assume A15:born x (+) born y c= D;
set CC =(comp(L_x,x,y,L_y) \/ comp(R_x,x,y,R_y)) \/
(comp(L_x,x,y,R_y) \/ comp(R_x,x,y,L_y));
A16:for X,Y be surreal-membered set st X c= L_x\/R_x & Y c=L_y\/R_y
holds comp(X,x,y,Y) is surreal-membered
proof
let X,Y be surreal-membered set such that
A17: X c= L_x\/R_x & Y c=L_y\/R_y;
let z be object;
assume z in comp(X,x,y,Y);
then consider x1,y1 be Surreal such that
A18:z = (x1*y) +' (x*y1) +' -' (x1*y1) & x1 in X & y1 in Y by Def14;
A19: born x1 (+) born y1 in born x1 (+) born y &
born x (+) born y1 in born x (+) born y &
born x1 (+) born y in born x (+) born y
by A17,SURREALO:1,A18,ORDINAL7:94;
then born x1 (+) born y1 in born x (+) born y by ORDINAL1:10;
then reconsider x1y=x1*y,xy1=x*y1,x1y1=x1*y1 as Surreal by A19,A15,A2;
thus thesis by A18;
end;
A20:L_x c= L_x\/R_x & R_x c= L_x\/R_x &
L_y c= L_y\/R_y & R_y c= L_y\/R_y by XBOOLE_1:7;
defpred P[object,object] means $1 is Surreal &
for z be Surreal st z = $1 holds $2 = born z;
A21: for x,y,z being object st P[x,y] & P[x,z] holds y = z
proof
let x,y,z be object such that A22: P[x,y] & P[x,z];
reconsider x as Surreal by A22;
thus y=born x by A22
.=z by A22;
end;
consider OO be set such that
A23: for z be object holds z in OO iff ex y be object st y in CC & P[y,z]
from TARSKI_0:sch 1(A21);
for x be set st x in OO holds x is ordinal
proof
let x be set;
assume x in OO;
then consider y be object such that
A24: y in CC & P[y,x] by A23;
reconsider y as Surreal by A24;
x= born y by A24;
hence thesis;
end;
then OO is ordinal-membered by ORDINAL6:1;
then reconsider U=union OO as Ordinal;
A25: for o be object st o in CC
ex O be Ordinal st O in succ U & o in Day O
proof
let o be object such that A26: o in CC;
A27: comp(L_x,x,y,L_y) is surreal-membered &
comp(R_x,x,y,R_y) is surreal-membered &
comp(L_x,x,y,R_y) is surreal-membered &
comp(R_x,x,y,L_y) is surreal-membered by A16,A20;
o in (comp(L_x,x,y,L_y) \/ comp(R_x,x,y,R_y))
or o in comp(L_x,x,y,R_y) \/ comp(R_x,x,y,L_y) by A26,XBOOLE_0:def 3;
then o in comp(L_x,x,y,L_y) or o in comp(R_x,x,y,R_y)
or o in comp(L_x,x,y,R_y) or o in comp(R_x,x,y,L_y) by XBOOLE_0:def 3;
then reconsider o as Surreal by A27;
P[o,born o];
then born o c= U by A23,A26,ZFMISC_1:74;
then A28: born o in succ U by ORDINAL1:6,12;
o in Day born o by SURREAL0:def 18;
hence thesis by A28;
end;
comp(L_x,x,y,L_y) \/ comp(R_x,x,y,R_y) <<
comp(L_x,x,y,R_y) \/ comp(R_x,x,y,L_y)
proof
let l,r be Surreal;
assume A29: l in comp(L_x,x,y,L_y) \/ comp(R_x,x,y,R_y) &
r in comp(L_x,x,y,R_y) \/ comp(R_x,x,y,L_y);
per cases by A29,XBOOLE_0:def 3;
suppose A30: l in comp(L_x,x,y,L_y) & r in comp(L_x,x,y,R_y);
then consider xL1,yL be Surreal such that
A31: l = (xL1*y) +' (x*yL) +' -' (xL1*yL) & xL1 in L_x & yL in L_y
by Def14;
consider xL2,yR be Surreal such that
A32: r = (xL2*y) +' (x*yR) +' -' (xL2*yR) & xL2 in L_x & yR in R_y
by Def14,A30;
xL1 in L_x \/ R_x & xL2 in L_x \/ R_x by A31,A32,XBOOLE_0:def 3;
then A33:born xL1 (+) born y in born x (+)born y &
born xL2 (+) born y in born x (+)born y by SURREALO:1,ORDINAL7:94;
then reconsider xL1y=xL1*y,xL2y=xL2*y as Surreal by A15,A2;
set BL = (born xL1 (+) born y) \/ (born xL2 (+) born y);
A34: BL in born x (+)born y by A33, ORDINAL3:12;
A35: born xL1 (+) born y c= BL & born xL2 (+) born y c= BL
by XBOOLE_1:7;
A36:xL1 == xL2 implies xL1y == xL2y by A35,A34,A15,A2;
A37:yR in L_y \/ R_y & yL in L_y \/ R_y by A31,A32,XBOOLE_0:def 3;
then A38:born xL1 (+) born yL in born xL1 (+)born y &
born xL1 (+) born yR in born xL1 (+)born y &
born xL2 (+) born yL in born xL2 (+)born y &
born xL2 (+) born yR in born xL2 (+)born y
by SURREALO:1,ORDINAL7:94;
then born xL1 (+) born yL in born x (+)born y &
born xL1 (+) born yR in born x (+)born y &
born xL2 (+) born yR in born x (+)born y &
born xL2 (+) born yL in born x (+)born y
by A33,ORDINAL1:10;
then reconsider xL1yR=xL1*yR,xL2yR=xL2*yR,xL2yL=xL2*yL,
xL1yL=xL1*yL as Surreal by A15,A2;
set BY=(born x (+) born yL)\/(born x (+) born yR);
A39: born x (+) born yL in born x (+)born y &
born x (+) born yR in born x (+)born y by
A37,SURREALO:1,ORDINAL7:94;
then reconsider xyL=x*yL,xyR=x*yR as Surreal by A15,A2;
A40: born xL1 (+) born yR c= BL & born xL2 (+) born yR c= BL &
born xL2 (+) born yL c= BL &
born xL1 (+) born yL c= BL by A38,A35,ORDINAL1:def 2;
A41:xL1 == xL2 implies xL1yR == xL2yR by A40,A34,A15,A2;
L_y << R_y by SURREAL0:45;
then A42: yL < yR by A31,A32;
x <=x & y <= y;
then A43: L_x << {x} << R_x & x in {x} &
L_y << {y} << R_y & y in {y} by SURREAL0:43,TARSKI:def 1;
then A44: xL1<x & xL2<x & yL < y by A31,A32;
A45: BY in born x (+)born y by A39,ORDINAL3:12;
A46: born x (+) born yL c= BY & born x (+) born yR c= BY
by XBOOLE_1:7;
set BB = BL \/BY;
A47: BB in born x (+)born y by A34,A45,ORDINAL3:12;
A48: BL c= BB & BY c= BB by XBOOLE_1:7;
then A49: born xL1 (+) born yL c= BB&
born xL1 (+) born yR c= BB &
born x (+) born yL c= BB & born x (+) born yR c= BB
by A40,A46,XBOOLE_1:1;
A50: born xL1 (+) born y c= BB & born xL2 (+) born y c= BB
by A35,A48,XBOOLE_1:1;
A51: born yL (+) born xL2 c= BB & born yR (+) born xL2 c= BB
by A40,A48,XBOOLE_1:1;
xL1yR+xyL < xL1yL+xyR by A47,A15,A2,A49,A42,A44;
then xyL < xL1yL+xyR - xL1yR by Th42;
then xyL < xL1yL+(xyR - xL1yR) by Th37;
then A52: xyL - xL1yL < xyR - xL1yR by Th41;
per cases;
suppose A53: xL1 < xL2;
xL2yL + xL1y < xL1yL + xL2y by A53,A44,A49,A50,A51,A47,A15,A2;
then xL1y < xL1yL + xL2y - xL2yL by Th42;
then xL1y < xL1yL +(xL2y - xL2yL) by Th37;
then xL1y - xL1yL < xL2y - xL2yL by Th41;
then A54: xL1y +- xL1yL + xyL <= xL2y - xL2yL + xyL by Th32;
A55: xL2y - xL2yL + xyL = xyL - xL2yL + xL2y by Th37;
xL2yR + xyL < xL2yL + xyR by A42,A44,A47,A15,A2,A49,A51;
then xyL < xL2yL + xyR - xL2yR by Th42;
then xyL < xL2yL + (xyR - xL2yR) by Th37;
then xyL - xL2yL < xyR - xL2yR by Th41;
then xyL - xL2yL + xL2y < xyR - xL2yR +xL2y by Th32;
then xL1y - xL1yL + xyL < xyR - xL2yR +xL2y
by SURREALO:4,A54,A55;
then xL1y +(- xL1yL + xyL) < xyR +- xL2yR +xL2y by Th37;
then xL1y +(xyL - xL1yL) < xyR +(xL2y- xL2yR) by Th37;
then xL1y +xyL - xL1yL < xyR +(xL2y- xL2yR) by Th37;
then (xL1y + xyL) + - (xL1yL) < (xL2y + xyR) + - xL2yR by Th37;
hence l < r by A31,A32;
end;
suppose A56: xL2 < xL1;
A57: born y (+) born xL1 c= BB & born yR (+) born xL1 c= BB
by A48,A40,XBOOLE_1:1,A35;
y < yR & xL2 < xL1 by A56,A43,A32;
then xL1y + xL2yR < xL2y + xL1yR by A50,A51,A57,A47,A15,A2;
then xL1y < xL2y + xL1yR -xL2yR by Th42;
then xL1y < xL1yR+ (xL2y -xL2yR) by Th37;
then xL1y - xL1yR < xL2y -xL2yR by Th41;
then xL1y - xL1yR +xyR < xL2y +-xL2yR +xyR by Th32;
then xL1y - xL1yR +xyR < xL2y +xyR+ -xL2yR by Th37;
then A58: xL1y + (xyR+- xL1yR) < xL2y +xyR+ -xL2yR by Th37;
xL1yR + xyL < xL1yL + xyR by A42,A44,A47,A15,A2,A49;
then xyL < xL1yL + xyR -xL1yR by Th42;
then xyL < xL1yL + (xyR -xL1yR) by Th37;
then xyL - xL1yL < xyR -xL1yR by Th41;
then xL1y +(xyL - xL1yL) <= xL1y +(xyR -xL1yR) by Th32;
then xL1y +(xyL - xL1yL) < xL2y +xyR-xL2yR by A58,SURREALO:4;
then (xL1y + xyL) + - (xL1yL) < (xL2y + xyR) - xL2yR by Th37;
hence l < r by A31,A32;
end;
suppose A59: xL1 == xL2;
then - xL1yR <= - xL2yR by A41,Th10;
then A60: (xL2y + xyR)- xL1yR <= (xL2y + xyR)- xL2yR by Th32;
xL1y + (xyL - xL1yL) < xL2y +(xyR - xL1yR) by A59,A36,A52,Th44;
then xL1y + (xyL - xL1yL) < (xL2y +xyR) - xL1yR by Th37;
then xL1y + (xyL - xL1yL) < (xL2y + xyR)- xL2yR
by A60,SURREALO:4;
then (xL1y + xyL) - (xL1yL) < (xL2y + xyR) - xL2yR by Th37;
hence l < r by A31,A32;
end;
end;
suppose A61: l in comp(R_x,x,y,R_y) & r in comp(R_x,x,y,L_y);
then consider xR1,yR be Surreal such that
A62: l = (xR1*y) +' (x*yR) +' -' (xR1*yR) & xR1 in R_x & yR in R_y
by Def14;
consider xR2,yL be Surreal such that
A63: r = (xR2*y) +' (x*yL) +' -' (xR2*yL) & xR2 in R_x & yL in L_y
by Def14,A61;
xR1 in L_x \/ R_x & xR2 in L_x \/ R_x by A62,A63,XBOOLE_0:def 3;
then A64:born xR1 (+) born y in born x (+)born y &
born xR2 (+) born y in born x (+)born y by SURREALO:1,ORDINAL7:94;
then reconsider xR1y=xR1*y,xR2y=xR2*y as Surreal by A15,A2;
set BR = (born xR1 (+) born y) \/ (born xR2 (+) born y);
A65: BR in born x (+)born y by A64, ORDINAL3:12;
A66: born xR1 (+) born y c= BR & born xR2 (+) born y c= BR
by XBOOLE_1:7;
A67:xR1 == xR2 implies xR1y == xR2y by A66,A65,A15,A2;
A68: yR in L_y \/ R_y & yL in L_y \/ R_y by A62,A63,XBOOLE_0:def 3;
then A69: born xR1 (+) born yL in born xR1 (+)born y &
born xR1 (+) born yR in born xR1 (+)born y &
born xR2 (+) born yL in born xR2 (+)born y &
born xR2 (+) born yR in born xR2 (+)born y
by SURREALO:1,ORDINAL7:94;
born xR1 (+) born yL in born x (+)born y &
born xR1 (+) born yR in born x (+)born y &
born xR2 (+) born yR in born x (+)born y &
born xR2 (+) born yL in born x (+)born y
by A69,A64,ORDINAL1:10;
then reconsider xR1yR=xR1*yR,xR2yR=xR2*yR,xR2yL=xR2*yL,xR1yL=xR1*yL
as Surreal by A15,A2;
set BY=(born x (+) born yL)\/(born x (+) born yR);
A70: born x (+) born yL in born x (+)born y &
born x (+) born yR in born x (+)born y
by A68,SURREALO:1,ORDINAL7:94;
then reconsider xyL=x*yL,xyR=x*yR as Surreal by A15,A2;
A71: born xR1 (+) born yR c= BR & born xR2 (+) born yR c= BR &
born xR2 (+) born yL c= BR &
born xR1 (+) born yL c= BR by A69, A66,ORDINAL1:def 2;
A72:xR1 == xR2 implies xR1yR == xR2yR by A71,A65, A15,A2;
L_y << R_y by SURREAL0:45;
then A73: yL < yR by A62,A63;
x <=x & y <= y;
then A74: L_x << {x} << R_x & x in {x} &
L_y << {y} << R_y & y in {y} by SURREAL0:43,TARSKI:def 1;
then A75: x<xR1 & x<xR2 & yL < y by A62,A63;
A76: BY in born x (+)born y by A70,ORDINAL3:12;
A77: born x (+) born yL c= BY & born x (+) born yR c= BY
by XBOOLE_1:7;
set BB = BR \/BY;
A78: BB in born x (+)born y by A65,A76,ORDINAL3:12;
A79:BR c= BB & BY c= BB by XBOOLE_1:7;
then
A80:born xR1 (+) born yL c= BB& born xR1 (+) born yR c= BB &
born x (+) born yL c= BB & born x (+) born yR c= BB
by A71,A77,XBOOLE_1:1;
A81: born xR1 (+) born y c= BB & born xR2 (+) born y c= BB
by A66,A79,XBOOLE_1:1;
A82: born xR2 (+) born yL c= BB &
born xR2 (+) born yR c= BB by A71,A79,XBOOLE_1:1;
per cases;
suppose A83: xR1 < xR2;
yL < y & xR1 < xR2 by A83,A74,A63;
then xR2yL +xR1y < xR1yL+xR2y by A80,A81,A82,A78,A15,A2;
then xR1y < xR1yL+xR2y - xR2yL by Th42;
then xR1y < xR1yL+(xR2y - xR2yL) by Th37;
then xR1y - xR1yL < xR2y - xR2yL by Th41;
then (xR1y - xR1yL)+xyL < xR2y - xR2yL +xyL by Th32;
then (- xR1yL + xR1y)+xyL < - xR2yL+ (xR2y +xyL) by Th37;
then A84: xR1y + xyL - xR1yL < xR2y + xyL - xR2yL by Th37;
xyR + xR1yL < xyL + xR1yR by A73,A75,A80,A78,A15,A2;
then xyR + xR1yL - xR1yR < xyL by Th41;
then xR1yL+ (xyR - xR1yR) < xyL by Th37;
then xyR +- xR1yR < xyL -xR1yL by Th42;
then xR1y + (xyR +- xR1yR) < xR1y+(xyL +-xR1yL) by Th32;
then (xR1y + xyR) +- xR1yR < xR1y+(xyL +-xR1yL) by Th37;
then xR1y + xyR + - xR1yR <= xR1y + xyL +- xR1yL by Th37;
hence thesis by A62,A63,A84,SURREALO:4;
end;
suppose A85: xR2 < xR1;
y < yR & xR2 < xR1 by A85,A74,A62;
then xR1y + xR2yR < xR2y + xR1yR by A80,A81,A82,A78,A15,A2;
then xR2yR +xR1y - xR1yR < xR2y by Th41;
then xR2yR +(xR1y - xR1yR) < xR2y by Th37;
then xR1y + - xR1yR < xR2y -xR2yR by Th42;
then xR1y + - xR1yR + xyR < xR2y + -xR2yR + xyR by Th32;
then A86: xR1y + xyR + - xR1yR < xR2y + -xR2yR + xyR by Th37;
xyR + xR2yL < xyL +xR2yR by A73,A75,A78,A15,A2,A80,A82;
then xyR < xR2yR + xyL - xR2yL by Th42;
then xyR < xR2yR + (xyL - xR2yL) by Th37;
then xyR -xR2yR < xyL - xR2yL by Th41;
then xR2y +(-xR2yR + xyR) < xR2y +(xyL + - xR2yL) by Th32;
then xR2y +-xR2yR + xyR < xR2y +(xyL + - xR2yL) by Th37;
then xR2y + -xR2yR + xyR <= xR2y + xyL + - xR2yL by Th37;
hence thesis by A62,A63,A86,SURREALO:4;
end;
suppose A87: xR1 == xR2;
then - xR1yR <= - xR2yR by A72,Th10;
then A88: (xR1y + xyR)+- xR1yR <= (xR1y + xyR)+- xR2yR by Th32;
xyR + xR2yL < xyL +xR2yR by A75,A73,A80,A82,A78,A15,A2;
then xR2yL + xyR - xR2yR < xyL by Th41;
then xR2yL + (xyR - xR2yR) < xyL by Th37;
then xyR - xR2yR < xyL - xR2yL by Th42;
then xR1y + (xyR - xR2yR) < xR2y+ (xyL - xR2yL) by A87,A67,Th44;
then xR1y + (xyR - xR2yR) < (xR2y+ xyL) - xR2yL by
Th37;
then xR1y + xyR +- xR2yR < (xR2y+ xyL) +- xR2yL by Th37;
hence l < r by A88,SURREALO:4,A62,A63;
end;
end;
suppose A89: l in comp(L_x,x,y,L_y) & r in comp(R_x,x,y,L_y);
then consider xL,yL1 be Surreal such that
A90: l = (xL*y) +' (x*yL1) +' -' (xL*yL1) & xL in L_x & yL1 in L_y
by Def14;
consider xR,yL2 be Surreal such that
A91: r = (xR*y) +' (x*yL2) +' -' (xR*yL2) & xR in R_x & yL2 in L_y
by Def14,A89;
yL1 in L_y \/ R_y & yL2 in L_y \/ R_y by A90,A91,XBOOLE_0:def 3;
then A92:born yL1 (+) born x in born x (+)born y &
born yL2 (+) born x in born x (+)born y by SURREALO:1,ORDINAL7:94;
then reconsider yL1x=yL1*x,yL2x=yL2*x as Surreal by A15,A2;
set BL = (born yL1 (+) born x) \/ (born yL2 (+) born x);
A93: BL in born x (+)born y by A92, ORDINAL3:12;
A94: born yL1 (+) born x c= BL & born yL2 (+) born x c= BL by
XBOOLE_1:7;
A95:yL1 == yL2 implies yL1x == yL2x by A94,A93,A15,A2;
A96: xR in L_x \/ R_x & xL in L_x \/ R_x by A90,A91,XBOOLE_0:def 3;
A97: born yL1 (+) born xL in born yL1 (+)born x &
born yL1 (+) born xR in born yL1 (+)born x &
born yL2 (+) born xL in born yL2 (+)born x &
born yL2 (+) born xR in born yL2 (+)born x
by A96,SURREALO:1,ORDINAL7:94;
then born yL1 (+) born xL in born x (+)born y &
born yL1 (+) born xR in born x (+)born y &
born yL2 (+) born xR in born x (+)born y &
born yL2 (+) born xL in born x (+)born y
by A92,ORDINAL1:10;
then reconsider yL1xR=yL1*xR,yL2xR=yL2*xR,yL2xL=yL2*xL,yL1xL=yL1*xL
as Surreal by A15,A2;
set BY=(born y (+) born xL)\/(born y (+) born xR);
A98: born y (+) born xL in born x (+)born y &
born y (+) born xR in born x (+)born y
by A96,SURREALO:1,ORDINAL7:94;
then reconsider yxL=y*xL,yxR=y*xR as Surreal by A15,A2;
A99: born yL1 (+) born xR c= BL & born yL2 (+) born xR c= BL &
born yL2 (+) born xL c= BL &
born yL1 (+) born xL c= BL by A97,A94,ORDINAL1:def 2;
A100:yL1 == yL2 implies yL1xR == yL2xR by A99,A93,A15,A2;
L_x << R_x by SURREAL0:45;
then A101: xL < xR by A90,A91;
x <=x & y <= y;
then A102: L_x << {x} << R_x & x in {x} &
L_y << {y} << R_y & y in {y} by SURREAL0:43,TARSKI:def 1;
then A103: yL1<y & yL2<y & xL < x by A90,A91;
A104: BY in born x (+)born y by A98,ORDINAL3:12;
A105: born y (+) born xL c= BY & born y (+) born xR c= BY
by XBOOLE_1:7;
set BB = BL \/BY;
A106: BB in born x (+)born y by A93,A104,ORDINAL3:12;
A107: BL c= BB & BY c= BB by XBOOLE_1:7;
then A108: born yL1 (+) born xL c= BB&
born yL1 (+) born xR c= BB &
born y (+) born xL c= BB & born y (+) born xR c= BB
by A99,A105,XBOOLE_1:1;
A109: born yL1 (+) born x c= BB & born yL2 (+) born x c= BB
by A94,A107,XBOOLE_1:1;
A110: born xL (+) born yL2 c= BB & born xR (+) born yL2 c= BB
by A99,A107,XBOOLE_1:1;
yL1xR+yxL < yL1xL+yxR by A106,A15,A2,A108,A101,A103;
then yxL < yL1xL+yxR - yL1xR by Th42;
then yxL < yL1xL+(yxR - yL1xR) by Th37;
then A111: yxL - yL1xL < yxR - yL1xR by Th41;
A112: xL*yL2 = yL2xL & x*yL2 = yL2x by A2,A106,A15,A109,A110;
A113: xL*yL1 = yL1xL & x*yL1 = yL1x by A2,A106,A15,A108,A109;
A114: xR*yL2 = yL2xR & xR*yL1 = yL1xR by A2,A106,A15,A108,A110;
A115: l = (yxL) + (yL1x) + - (yL1xL) by A90,A2,A106,A15,A108,A113
.= yL1x + yxL +- (yL1xL);
A116: r = yxR + yL2x + - yL2xR by A91,A2,A106,A15,A108,A112,A114
.= yL2x + yxR + - yL2xR;
per cases;
suppose A117: yL1 < yL2;
yL2xL + yL1x < yL1xL + yL2x
by A117,A103,A108,A109,A110,A106,A15,A2;
then yL1x < yL1xL + yL2x - yL2xL by Th42;
then yL1x < yL1xL +(yL2x - yL2xL) by Th37;
then yL1x - yL1xL < yL2x - yL2xL by Th41;
then A118: yL1x +- yL1xL + yxL <= yL2x - yL2xL + yxL by Th32;
A119: yL2x - yL2xL + yxL =yxL - yL2xL + yL2x by Th37;
yL2xR + yxL < yL2xL + yxR by A101,A103,A106,A15,A2,A108,A110;
then yxL < yL2xL + yxR - yL2xR by Th42;
then yxL < yL2xL + (yxR +- yL2xR) by Th37;
then yxL - yL2xL < yxR +- yL2xR by Th41;
then yxL - yL2xL + yL2x < yxR +- yL2xR +yL2x by Th32;
then yL1x - yL1xL + yxL < yxR +- yL2xR +yL2x
by SURREALO:4,A118,A119;
then yL1x +(- yL1xL + yxL) < yxR - yL2xR +yL2x by Th37;
then yL1x +(yxL - yL1xL) < yxR +(yL2x- yL2xR) by Th37;
then yL1x +yxL - yL1xL < yxR +(yL2x- yL2xR) by Th37;
hence l < r by A115,A116,Th37;
end;
suppose A120: yL2 < yL1;
A121: born x (+) born yL1 c= BB & born xR (+) born yL1 c= BB
by A99,XBOOLE_1:1,A94,A107;
x < xR & yL2 < yL1 by A120,A102,A91;
then yL1x + yL2xR < yL2x + yL1xR by A109,A110,A121,A106,A15,A2;
then yL1x < yL2x + yL1xR -yL2xR by Th42;
then yL1x < yL1xR+ (yL2x -yL2xR) by Th37;
then yL1x - yL1xR < yL2x -yL2xR by Th41;
then yL1x - yL1xR +yxR < yL2x -yL2xR +yxR by Th32;
then yL1x - yL1xR +yxR < yL2x +yxR -yL2xR by Th37;
then A122:yL1x + (yxR+- yL1xR) < yL2x +yxR+ -yL2xR by Th37;
yL1xR + yxL < yL1xL + yxR by A101,A103,A106,A15,A2,A108;
then yxL < yL1xL + yxR -yL1xR by Th42;
then yxL < yL1xL + (yxR -yL1xR) by Th37;
then yxL - yL1xL < yxR -yL1xR by Th41;
then yL1x +(yxL +- yL1xL) <= yL1x +(yxR -yL1xR) by Th32;
then yL1x +(yxL +- yL1xL) < yL2x +yxR+ -yL2xR
by A122,SURREALO:4;
hence l < r by A115,A116,Th37;
end;
suppose A123: yL1 == yL2;
then - yL1xR <= - yL2xR by Th10,A100;
then A124: (yL2x + yxR)+- yL1xR <= (yL2x + yxR)+- yL2xR by Th32;
yL1x + (yxL +- yL1xL) < yL2x+(yxR +- yL1xR) by A123,A95,A111,Th44;
then yL1x + (yxL +- yL1xL) < (yL2x +yxR) +- yL1xR by Th37;
then yL1x + (yxL +- yL1xL) < (yL2x + yxR)+- yL2xR
by A124,SURREALO:4;
hence l < r by Th37,A115,A116;
end;
end;
suppose A125: l in comp(R_x,x,y,R_y) & r in comp(L_x,x,y,R_y);
then consider xR,yR1 be Surreal such that
A126: l = (xR*y) +' (x*yR1) +' -' (xR*yR1) & xR in R_x & yR1 in R_y
by Def14;
consider xL,yR2 be Surreal such that
A127: r = (xL*y) +' (x*yR2) +' -' (xL*yR2) & xL in L_x & yR2 in R_y
by Def14,A125;
yR1 in L_y \/ R_y & yR2 in L_y \/ R_y by A126,A127,XBOOLE_0:def 3;
then A128:born yR1 (+) born x in born y (+)born x &
born yR2 (+) born x in born y (+)born x by SURREALO:1,ORDINAL7:94;
then reconsider yR1x=yR1*x,yR2x=yR2*x as Surreal by A15,A2;
set BR = (born yR1 (+) born x) \/ (born yR2 (+) born x);
A129: BR in born x (+)born y by A128, ORDINAL3:12;
A130: born yR1 (+) born x c= BR & born yR2 (+) born x c= BR
by XBOOLE_1:7;
A131:yR1 == yR2 implies yR1x == yR2x by A130,A129,A15,A2;
A132: xR in L_x \/ R_x & xL in L_x \/ R_x
by A126,A127,XBOOLE_0:def 3;
A133: born yR1 (+) born xL in born yR1 (+)born x &
born yR1 (+) born xR in born yR1 (+)born x &
born yR2 (+) born xL in born yR2 (+)born x &
born yR2 (+) born xR in born yR2 (+)born x
by A132,SURREALO:1,ORDINAL7:94;
born yR1 (+) born xL in born x (+)born y &
born yR1 (+) born xR in born x (+)born y &
born yR2 (+) born xR in born x (+)born y &
born yR2 (+) born xL in born x (+)born y
by A128,ORDINAL1:10,A133;
then reconsider yR1xR=yR1*xR,yR2xR=yR2*xR,yR2xL=yR2*xL,
yR1xL=yR1*xL as Surreal by A15,A2;
set BY=(born y (+) born xL)\/(born y (+) born xR);
A134: born y (+) born xL in born x (+)born y &
born y (+) born xR in born x (+)born y
by A132,SURREALO:1,ORDINAL7:94;
then reconsider yxL=y*xL,yxR=y*xR as Surreal by A15,A2;
A135: born yR1 (+) born xR c= BR & born yR2 (+) born xR c= BR &
born yR2 (+) born xL c= BR &
born yR1 (+) born xL c= BR by A133,A130,ORDINAL1:def 2;
A136:yR1 == yR2 implies yR1xR == yR2xR by A135,A129,A15,A2;
L_x << R_x by SURREAL0:45;
then A137: xL < xR by A126,A127;
x <=x & y <= y;
then A138: L_x << {x} << R_x & x in {x} &
L_y << {y} << R_y & y in {y} by SURREAL0:43,TARSKI:def 1;
then
A139: y<yR1 & y<yR2 & xL < x by A126,A127;
A140: BY in born x (+)born y by A134,ORDINAL3:12;
A141: born y (+) born xL c= BY & born y (+) born xR c= BY
by XBOOLE_1:7;
set BB = BR \/BY;
A142:BB in born x (+)born y by A129,A140,ORDINAL3:12;
A143:BR c= BB & BY c= BB by XBOOLE_1:7;
then
A144: born yR1 (+) born xL c= BB&
born yR1 (+) born xR c= BB &
born y (+) born xL c= BB &
born y (+) born xR c= BB by A135,A141,XBOOLE_1:1;
A145: born yR1 (+) born x c= BB & born yR2 (+) born x c= BB
by A130,A143,XBOOLE_1:1;
A146: born yR2 (+) born xL c= BB &
born yR2 (+) born xR c= BB by A135,A143,XBOOLE_1:1;
A147: xR*y = yxR & xL*y = yxL by A2,A142,A15,A144;
A148: xL*yR2 = yR2xL & x*yR2 = yR2x by A2,A142,A15,A145,A146;
A149: xR*yR2 = yR2xR & xR*yR1 = yR1xR by A2,A142,A15,A144,A146;
A150: l = yxR + yR1x + - yR1xR by A126,A147,A2,A142,A15,A145,A149
.= yR1x +yxR +-yR1xR;
A151: r =yxL + yR2x + - yR2xL by A127,A2,A142,A15,A144,A148
.= yR2x + yxL + - yR2xL;
per cases;
suppose A152: yR1 < yR2;
yR2xL +yR1x < yR1xL+yR2x by A152,A139,A144,A145,A146,A142,A15,A2;
then yR1x < yR1xL+yR2x - yR2xL by Th42;
then yR1x < yR1xL+(yR2x - yR2xL) by Th37;
then yR1x - yR1xL < yR2x - yR2xL by Th41;
then (yR1x - yR1xL)+yxL < yR2x - yR2xL +yxL by Th32;
then (- yR1xL + yR1x)+yxL < - yR2xL+ (yR2x +yxL) by Th37;
then A153: yR1x + yxL +- yR1xL < yR2x + yxL - yR2xL by Th37;
yxR + yR1xL < yxL + yR1xR by A137,A139,A144,A142,A15,A2;
then yxR + yR1xL - yR1xR < yxL by Th41;
then yR1xL+ (yxR - yR1xR) < yxL by Th37;
then yxR +- yR1xR < yxL -yR1xL by Th42;
then yR1x + (yxR - yR1xR) < yR1x+(yxL -yR1xL) by Th32;
then (yR1x + yxR) - yR1xR < yR1x+(yxL -yR1xL) by Th37;
then yR1x + yxR - yR1xR <= yR1x + yxL - yR1xL by Th37;
hence thesis by A151,A150,A153,SURREALO:4;
end;
suppose A154: yR2 < yR1;
x < xR & yR2 < yR1 by A154,A138,A126;
then yR1x + yR2xR < yR2x + yR1xR by
A144,A145,A146,A142,A15,A2;
then yR1x + yR2xR - yR1xR < yR2x by Th41;
then yR2xR +(yR1x - yR1xR) < yR2x by Th37;
then yR1x - yR1xR < yR2x -yR2xR by Th42;
then yR1x - yR1xR + yxR < yR2x -yR2xR + yxR by Th32;
then A155: yR1x + yxR + - yR1xR < yR2x -yR2xR + yxR by Th37;
yxR + yR2xL < yxL +yR2xR by A137,A139,A142,A15,A2,A144,A146;
then yxR < yR2xR + yxL - yR2xL by Th42;
then yxR < yR2xR + (yxL - yR2xL) by Th37;
then yxR-yR2xR < yxL - yR2xL by Th41;
then yR2x +(-yR2xR + yxR) < yR2x +(yxL - yR2xL) by Th32;
then yR2x +-yR2xR + yxR < yR2x +(yxL - yR2xL) by Th37;
then yR2x -yR2xR + yxR <= yR2x + yxL - yR2xL by Th37;
hence thesis by A151,A150,A155,SURREALO:4;
end;
suppose A156: yR1 == yR2;
then - yR1xR <= - yR2xR by A136,Th10;
then A157: (yR1x + yxR)- yR1xR <= (yR1x + yxR)- yR2xR by Th32;
yxR + yR2xL < yxL +yR2xR by A139,A137,A144,A146,A142,A15,A2;
then yR2xL + yxR - yR2xR < yxL by Th41;
then yR2xL + (yxR - yR2xR) < yxL by Th37;
then yxR - yR2xR < yxL - yR2xL by Th42;
then yR1x + (yxR - yR2xR) < yR2x+ (yxL - yR2xL)
by A156,A131,Th44;
then yR1x + (yxR - yR2xR) < (yR2x+ yxL) - yR2xL by Th37;
then yR1x + yxR - yR2xR < (yR2x+ yxL) - yR2xL by Th37;
hence l < r by A151,A150,A157,SURREALO:4;
end;
end;
end;
then [(comp(L_x,x,y,L_y) \/ comp(R_x,x,y,R_y)),
(comp(L_x,x,y,R_y) \/ comp(R_x,x,y,L_y))] in Day succ U
by A25,SURREAL0:46;
hence thesis by Th50;
end;
defpred Q24[Ordinal] means
for x1,x2,y be Surreal st born x1 (+) born x2 (+) born y c= $1
holds P2[D,x1,x2,y] & P4[D,x1,x2,y];
A158: for E be Ordinal st for C be Ordinal st C in E holds Q24[C]
holds Q24[E]
proof
let E be Ordinal such that A159: for C be Ordinal st C in E holds Q24[C];
A160: for x1,x2,y,x1y,x2y be Surreal st
born x1 (+) born x2 (+) born y c= E &
born x1 (+) born y c= D & born x2 (+) born y c= D &
x1 == x2 & x1y = x1*y & x2y = x2*y
holds L_x1y << {x2y} << R_x1y
proof
let x1,x2,y,x1y,x2y be Surreal such that
A161: born x1 (+) born x2 (+) born y c= E
and
A162: born x1 (+) born y c= D & born x2 (+) born y c= D and
A163: x1 == x2 & x1y = x1*y & x2y = x2*y;
A164:x1y =[(comp(L_x1,x1,y,L_y) \/ comp(R_x1,x1,y,R_y)),
(comp(L_x1,x1,y,R_y) \/ comp(R_x1,x1,y,L_y))] by A163,Th50;
A165: L_y << {y} << R_y & y in {y} by SURREALO:11,TARSKI:def 1;
thus L_x1y << {x2y}
proof
let l,r such that A166: l in L_x1y & r in {x2y};
per cases by A164,A166,XBOOLE_0:def 3;
suppose
l in comp(R_x1,x1,y,R_y);
then consider x3,y3 be Surreal such that
A167: l = (x3*y) +' (x1*y3) +' -' (x3*y3) & x3 in R_x1 & y3 in R_y
by Def14;
A168: {x1} << R_x1 & x1 in {x1} by SURREALO:11,TARSKI:def 1;
A169: y3 in L_y \/ R_y by A167,XBOOLE_0:def 3;
then A170:born x1 (+) born y3 in born x1 (+)born y &
born x2 (+) born y3 in born x2 (+)born y
by SURREALO:1,ORDINAL7:94;
then reconsider x1y3=x1*y3,x2y3=x2*y3 as Surreal by A2,A162;
set BL = (born x1 (+) born y3) \/ (born x2 (+) born y3);
x3 in L_x1 \/R_x1 by A167,XBOOLE_0:def 3;
then A171: born x3 (+) born y3 in born x1 (+) born y3 &
born x3 (+) born y in born x1 (+) born y
by SURREALO:1,ORDINAL7:94;
then A172: born x3 (+) born y3 in D &
born x3 (+) born y in D by A170,A162,ORDINAL1:10;
then reconsider x3y3=x3*y3,x3y=x3*y as Surreal by A2;
A173: (born x1 (+) born x2) (+) born y3 in
(born x1 (+) born x2) (+) born y
by A169,SURREALO:1,ORDINAL7:94;
born x1 (+) born y3 c= D & born x2 (+) born y3 c= D
by A170,A162,ORDINAL1:def 2;
then A174: x1y3 == x2y3 by A173,A163,A161,A159;
born y (+) born x3 (+) born x2 in born y (+) born x1 (+) born x2
by A171,ORDINAL7:94;
then A175: born y (+) born x3 (+) born x2 in
born y (+) (born x1 (+) born x2) by ORDINAL7:68;
A176: born y (+) born x3 (+) born x2 =
born y (+) born x2 (+) born x3 by ORDINAL7:68;
A177:born x2 (+) born y3 c= D by A170,A162,ORDINAL1:def 2;
A178: y3*x2 = x2y3 by A3,A170,A162,ORDINAL1:def 2;
A179:born x3 (+) born y c= D & born x2 (+) born y c= D
by A171,A162,ORDINAL1:def 2;
A180: y*x3 = x3y & y*x2 = x2y by A162,A163,A3,A171,A2;
A181:born x3 (+) born y3 c= D
by A171,A170,A162,ORDINAL1:10,ORDINAL1:def 2;
A182: y3*x3 = x3y3 by A172,A2;
y < y3 & x2 < x3 by A168,A167,A163,SURREALO:4,A165;
then x2y3 + x3y < x3y3 + x2y
by A167,A176,A175,A161,A159,A177,A178,A179,A180,A181,A182;
then A183: x2y3 + x3y -x3y3 < x2y by Th41;
x1y3 +(x3y +-x3y3) <= x2y3 +(x3y -x3y3) by A174,Th32;
then x1y3 +x3y -x3y3 <= x2y3 +(x3y -x3y3) by Th37;
then x1y3 +x3y -x3y3 <= x2y3 +x3y -x3y3 by Th37;
then A184: x1y3 +x3y -x3y3 < x2y by A183,SURREALO:4;
l = x3y+x1y3-x3y3 by A167;
hence thesis by A166,A184,TARSKI:def 1;
end;
suppose l in comp(L_x1,x1,y,L_y);
then consider x3,y3 be Surreal such that
A185: l = (x3*y) +' (x1*y3) +' -' (x3*y3) & x3 in L_x1 & y3 in L_y
by Def14;
A186: L_x1 << {x1} & x1 in {x1} by SURREALO:11,TARSKI:def 1;
A187: y3 in L_y \/ R_y by A185,XBOOLE_0:def 3;
then A188:born x1 (+) born y3 in born x1 (+)born y &
born x2 (+) born y3 in born x2 (+)born y
by SURREALO:1,ORDINAL7:94;
then reconsider x1y3=x1*y3,x2y3=x2*y3 as Surreal by A2,A162;
set BL = (born x1 (+) born y3) \/ (born x2 (+) born y3);
x3 in L_x1 \/R_x1 by A185,XBOOLE_0:def 3;
then A189: born x3 (+) born y3 in born x1 (+) born y3 &
born x3 (+) born y in born x1 (+) born y
by SURREALO:1,ORDINAL7:94;
then A190: born x3 (+) born y3 in D & born x3 (+) born y in D
by A188,A162,ORDINAL1:10;
then reconsider x3y3=x3*y3,x3y=x3*y as Surreal by A2;
A191: born x1 (+) born x2 (+) born y3 in
born x1 (+) born x2 (+) born y
by A187,SURREALO:1,ORDINAL7:94;
born x1 (+) born y3 c= D & born x2 (+) born y3 c= D
by A188,A162,ORDINAL1:def 2;
then A192: x1y3 == x2y3 by A191,A161,A159,A163;
born y (+) born x3 (+) born x2 in born y (+) born x1 (+) born x2
by A189,ORDINAL7:94;
then A193: born y (+) born x3 (+) born x2 in
born y (+) (born x1 (+) born x2) by ORDINAL7:68;
A194:born x2 (+) born y3 c= D by A188,A162,ORDINAL1:def 2;
A195: y3*x2 = x2y3 by A3,A188,A162,ORDINAL1:def 2;
A196:born x3 (+) born y c= D & born x2 (+) born y c= D
by A189,A162,ORDINAL1:def 2;
A197: y*x3 = x3y & y*x2 = x2y
by A163,A3,A162,A189,ORDINAL1:def 2;
A198:born x3 (+) born y3 c= D
by A188,A162,ORDINAL1:10,A189,ORDINAL1:def 2;
A199: y3*x3 = x3y3 by A190,A2;
A200: y3<y & x3 < x2 by A186,A185,A163,SURREALO:4,A165;
x2y3 + x3y < x3y3 + x2y
by A185,A200,A194,A195,A196,A197,A198,A199,A193,A161,A159;
then A201: x2y3 + x3y -x3y3 < x2y by Th41;
x1y3 +(x3y +-x3y3) <= x2y3 +(x3y +-x3y3) by A192,Th32;
then x1y3 +x3y +-x3y3 <= x2y3 +(x3y +-x3y3) by Th37;
then x1y3 +x3y +-x3y3 <= x2y3 +x3y +-x3y3 by Th37;
then A202: x1y3 +x3y +-x3y3 < x2y by A201,SURREALO:4;
l = x3y+x1y3+-x3y3 by A185;
hence thesis by A166,A202,TARSKI:def 1;
end;
end;
thus {x2y} << R_x1y
proof
let l,r such that A203: l in {x2y} & r in R_x1y;
per cases by XBOOLE_0:def 3,A164,A203;
suppose
r in comp(L_x1,x1,y,R_y);
then consider x3,y3 be Surreal such that
A204: r = (x3*y) +' (x1*y3) +' -' (x3*y3) &
x3 in L_x1 & y3 in R_y by Def14;
A205: L_x1 << {x1} & x1 in {x1} by SURREALO:11,TARSKI:def 1;
A206:y3 in L_y \/ R_y by A204,XBOOLE_0:def 3;
then A207:born x1 (+) born y3 in born x1 (+)born y &
born x2 (+) born y3 in born x2 (+)born y
by SURREALO:1,ORDINAL7:94;
then reconsider x1y3=x1*y3,x2y3=x2*y3 as Surreal by A162,A2;
set BL = (born x1 (+) born y3) \/ (born x2 (+) born y3);
x3 in L_x1 \/R_x1 by A204,XBOOLE_0:def 3;
then A208: born x3 (+) born y3 in born x1 (+) born y3 &
born x3 (+) born y in born x1 (+) born y
by SURREALO:1,ORDINAL7:94;
then A209:born x3 (+) born y3 in D &
born x3 (+) born y in D
by A207,A162,ORDINAL1:10;
then reconsider x3y3=x3*y3,x3y=x3*y as Surreal by A2;
A210: born x1 (+) born x2 (+) born y3 in
born x1 (+) born x2 (+) born y
by A206,SURREALO:1,ORDINAL7:94;
born x1 (+) born y3 c= D & born x2 (+) born y3 c= D
by A207,A162,ORDINAL1:def 2;
then A211: x1y3 == x2y3 by A210,A161,A159,A163;
born y (+) born x3 (+) born x2 in born y (+) born x1 (+) born x2
by A208,ORDINAL7:94;
then A212: born y (+) born x3 (+) born x2 in
born y (+) (born x1 (+) born x2) by ORDINAL7:68;
A213:born x2 (+) born y3 c= D by A207,A162,ORDINAL1:def 2;
A214: y3*x2 = x2y3 by A3,A207,A162,ORDINAL1:def 2;
A215:born x3 (+) born y c= D & born x2 (+) born y c= D
by A208,A162,ORDINAL1:def 2;
A216: y*x3 = x3y & y*x2 = x2y by A208,A162,ORDINAL1:def 2,A163,A3;
A217:born x3 (+) born y3 c= D
by A207,A162,ORDINAL1:10, A208,ORDINAL1:def 2;
A218: y3*x3 = x3y3 by A209,A2;
A219: y < y3 & x3 < x2 by A205,A204,A163,SURREALO:4,A165;
x3y3 + x2y < x2y3 + x3y
by A204,A212,A161,A159,A219,A213,A214,A215,A216,A217,A218;
then A220: x2y < x2y3 + x3y -x3y3 by Th42;
x2y3 +(x3y +-x3y3) <= x1y3 +(x3y +-x3y3) by A211,Th32;
then x2y3 +x3y +-x3y3 <= x1y3 +(x3y +-x3y3) by Th37;
then x2y3 +x3y +-x3y3 <= x1y3 +x3y +-x3y3 by Th37;
then A221: x2y < x1y3 +x3y +-x3y3 by A220,SURREALO:4;
r = x3y+x1y3+-x3y3 by A204;
hence thesis by A203,A221,TARSKI:def 1;
end;
suppose r in comp(R_x1,x1,y,L_y);
then consider x3,y3 be Surreal such that
A222: r = (x3*y) +' (x1*y3) +' -' (x3*y3) & x3 in R_x1 & y3 in L_y
by Def14;
A223: {x1} << R_x1 &
x1 in {x1} by SURREALO:11,TARSKI:def 1;
A224: y3 in L_y \/ R_y by A222,XBOOLE_0:def 3;
then A225:born x1 (+) born y3 in born x1 (+)born y &
born x2 (+) born y3 in born x2 (+)born y
by SURREALO:1,ORDINAL7:94;
then reconsider x1y3=x1*y3,x2y3=x2*y3 as Surreal by A2,A162;
set BL = (born x1 (+) born y3) \/ (born x2 (+) born y3);
x3 in L_x1 \/R_x1 by A222,XBOOLE_0:def 3;
then A226: born x3 (+) born y3 in born x1 (+) born y3 &
born x3 (+) born y in born x1 (+) born y
by SURREALO:1,ORDINAL7:94;
then A227:born x3 (+) born y3 in D & born x3 (+) born y in D
by A225,A162,ORDINAL1:10;
then reconsider x3y3=x3*y3,x3y=x3*y as Surreal by A2;
A228:(born x1 (+) born x2) (+) born y3 in
(born x1 (+) born x2) (+) born y
by A224,SURREALO:1,ORDINAL7:94;
born x1 (+) born y3 c= D & born x2 (+) born y3 c= D
by A225,A162,ORDINAL1:def 2;
then A229: x1y3 == x2y3 by A228,A159,A161,A163;
born y (+) born x3 (+) born x2 in born y (+) born x1 (+) born x2
by A226,ORDINAL7:94;
then A230: born y (+) born x3 (+) born x2 in
born y (+) (born x1 (+) born x2) by ORDINAL7:68;
A231: born y (+) born x3 (+) born x2
= born y (+) born x2 (+) born x3 by ORDINAL7:68;
A232:born x2 (+) born y3 c= D by A225,A162,ORDINAL1:def 2;
A233: y3*x2 = x2y3 by A3,A225,A162,ORDINAL1:def 2;
A234:born x3 (+) born y c= D & born x2 (+) born y c= D
by A226,A162,ORDINAL1:def 2;
A235: y*x3 = x3y & y*x2 = x2y by A163,A3,A226,A162,ORDINAL1:def 2;
A236:born x3 (+) born y3 c= D by A226,A225,A162,ORDINAL1:10,
ORDINAL1:def 2;
A237: y3*x3 = x3y3 by A3,A227,ORDINAL1:def 2;
A238: y3<y & x2 < x3 by A223,A222,A163,SURREALO:4,A165;
x3y3 + x2y < x2y3 + x3y
by A231,A222,A230,A159,A161,A238,A232,A233,A234,A235,A236,A237;
then A239: x2y < x2y3 + x3y -x3y3 by Th42;
x2y3 +(x3y +-x3y3) <= x1y3 +(x3y -x3y3) by A229,Th32;
then x2y3 +x3y -x3y3 <= x1y3 +(x3y -x3y3) by Th37;
then x2y3 +x3y -x3y3 <= x1y3 +x3y -x3y3 by Th37;
then A240: x2y < x1y3 +x3y -x3y3 by A239,SURREALO:4;
r = x3y+x1y3-x3y3 by A222;
hence thesis by A203,A240,TARSKI:def 1;
end;
end;
end;
let x,y1,y2 be Surreal such that
A241: born x (+) born y1 (+) born y2 c= E;
thus P2[D,x,y1,y2]
proof
let x1y,x2y be Surreal such that
A242: born x (+) born y2 c= D & born y1 (+) born y2 c= D &
x == y1 & x1y = x*y2 & x2y = y1*y2;
A243: L_x1y << {x2y} << R_x1y & L_x2y << {x1y} << R_x2y
by A241,A242,A160;
then reconsider z= [L_x1y \/ L_x2y, R_x1y \/ R_x2y] as Surreal
by SURREALO:14;
x1y == z == x2y by A243,SURREALO:15;
hence x1y == x2y by SURREALO:4;
end;
assume y1 < y2;
then per cases by SURREALO:13;
suppose ex y2L be Surreal st y2L in L_y2 & y1 <= y2L < y2;
then consider y2L be Surreal such that
A244: y2L in L_y2 & y1 <= y2L < y2;
A245:for xL be Surreal st xL in L_x holds P3[D,xL,x,y1,y2]
proof
let xL be Surreal such that A246: xL in L_x;
let xLy2,xy1,xLy1,xy2 be Surreal such that
A247:born xL (+) born y1 c= D & born x (+) born y1 c= D &
born xL (+) born y2 c= D & born x (+) born y2 c= D and
A248: xLy1 = xL*y1 & xLy2=xL*y2 & xy1=x*y1 & xy2=x*y2 and
A249: xL < x & y1 < y2;
A250: xy2 = [comp(L_x,x,y2,L_y2) \/ comp(R_x,x,y2,R_y2),
comp(L_x,x,y2,R_y2) \/ comp(R_x,x,y2,L_y2)] by A248,Th50;
A251: L_xy2 << {xy2} << R_xy2 & xy2 in {xy2}
by SURREALO:11,TARSKI:def 1;
A252: y2L in L_y2 \/ R_y2 by A244,XBOOLE_0:def 3;
then A253: born x (+) born y2L in born x (+) born y2
by SURREALO:1,ORDINAL7:94;
then reconsider xy2L=x*y2L as Surreal by A2,A247;
A254: xL in L_x \/ R_x by A246,XBOOLE_0:def 3;
then A255:born xL (+) born y2L in born x (+) born y2L
by SURREALO:1,ORDINAL7:94;
then A256: born xL (+) born y2L in D by A247,A253,ORDINAL1:10;
then reconsider xLy2L=xL*y2L as Surreal by A2;
(xLy2) +' xy2L +' -' xLy2L in comp(L_x,x,y2,L_y2)
by A246,A244,A248,Def14;
then (xLy2) +' xy2L +' -' xLy2L in L_xy2 by A250,XBOOLE_0:def 3;
then xLy2 + xy2L - xLy2L < xy2 by A251;
then A257: xLy2 + xy2L < xy2 + xLy2L by Th41;
A258: born y1 (+) born y2 (+) born x = born x (+) born y1 (+) born y2
by ORDINAL7:68;
A259: born y2L (+) born x c= D by A253,A247,ORDINAL1:def 2;
A260: y2L*x = xy2L & y1*x = xy1
by A253,A247,A248,A3,ORDINAL1:def 2;
A261: born y2L (+) born xL c= D
by A255,A247,A253,ORDINAL1:10,ORDINAL1:def 2;
A262: y2L*xL = xLy2L by A3,A256,ORDINAL1:def 2;
A263: y1*xL =xLy1 by A247,A248,A3;
A264: born y2L (+) born y1 in born y1 (+) born y2
by A252,SURREALO:1,ORDINAL7:94;
then A265: born y2L (+) born y1 (+) born x in
born y2 (+) born y1 (+) born x by ORDINAL7:94;
then A266: born y2L (+) born y1 (+) born x in E by A241,A258;
born y2L (+) born y1 (+) born xL in born y2 (+) born y1 (+) born xL &
born y2 (+) born y1 (+) born xL in
born y2 (+) born y1 (+) born x by A254,SURREALO:1,A264,ORDINAL7:94;
then
A267: born y2L (+) born y1 (+) born xL in
born y1 (+) born y2 (+) born x by ORDINAL1:10;
per cases by A244;
suppose A268:y1 < y2L;
born x (+) born y1 (+) born y2L in E by A266,ORDINAL7:68;
then A269: xLy2L+xy1 <= xLy1+xy2L
by A246,A159,A249,A268,A261,A247,A248,A259;
A270: xLy2L+xy1 +(xLy2 + xy2L) < xLy1+xy2L +(xy2 + xLy2L)
by A257,A269,Th44;
A271: xLy2L+xy1 +(xLy2 + xy2L) = xLy2L+(xy1 +(xLy2 + xy2L))
by Th37
.= xLy2L+((xy1 +xLy2) + xy2L) by Th37
.= (xLy2L + xy2L) + (xLy2+ xy1) by Th37;
xLy1+xy2L +(xy2 + xLy2L) = xy2L + (xLy1 +(xy2 + xLy2L)) by Th37
.= xy2L + ((xLy1 +xy2) + xLy2L) by Th37
.= (xLy2L + xy2L) + (xLy1+xy2) by Th37;
hence xLy2+ xy1 < xLy1+xy2 by A270,A271,Th43;
end;
suppose A272: y1 == y2L;
A273: xy2L == xy1 by A241,A258,A265,A159,A247,A259,A272,A260;
A274: xLy2L == xLy1
by A261,A263,A262,A247,A272,A267,A159,A241,A258;
A275: xy2 + xLy2L <= xLy1+xy2 by A274,Th32;
A276:xLy2 + xy1 <= xLy2 + xy2L by A273,Th32;
xLy2 + xy1 < xy2 + xLy2L by A257,A276,SURREALO:4;
hence xLy2+ xy1 < xLy1+xy2 by A275,SURREALO:4;
end;
end;
for xR be Surreal st xR in R_x holds P3[D,x,xR,y1,y2]
proof
let xR be Surreal such that A277: xR in R_x;
let xy2,xRy1,xy1,xRy2 be Surreal such that
A278:born x (+) born y1 c= D & born xR (+) born y1 c= D &
born x (+) born y2 c= D & born xR (+) born y2 c= D and
A279: xy1 = x*y1 & xy2=x*y2 & xRy1=xR*y1 & xRy2=xR*y2 and
A280: x < xR & y1 < y2;
A281: xy2 = [comp(L_x,x,y2,L_y2) \/ comp(R_x,x,y2,R_y2),
comp(L_x,x,y2,R_y2) \/ comp(R_x,x,y2,L_y2)] by A279,Th50;
A282: L_xy2 << {xy2} << R_xy2 & xy2 in {xy2}
by SURREALO:11,TARSKI:def 1;
A283: y2L in L_y2 \/ R_y2 by A244,XBOOLE_0:def 3;
A284: born x (+) born y2L in born x (+) born y2
by A283,SURREALO:1,ORDINAL7:94;
reconsider xy2L=x*y2L as Surreal by A2,A284,A278;
A285: xR in L_x \/ R_x by A277,XBOOLE_0:def 3;
A286: born xR (+) born y2L in born x (+) born y2L
by A285,SURREALO:1,ORDINAL7:94;
then A287: born xR (+) born y2L in D by A278,A284,ORDINAL1:10;
then reconsider xRy2L=xR*y2L as Surreal by A2;
(xRy2) +' xy2L +' -' xRy2L in comp(R_x,x,y2,L_y2)
by A277,A244,A279,Def14;
then (xRy2) +' xy2L +' -' xRy2L in R_xy2 by A281,XBOOLE_0:def 3;
then A288: xy2 < (xRy2) + xy2L - xRy2L by A282;
then A289: xy2 + xRy2L < (xRy2) + xy2L by Th42;
A290: born y1 (+) born y2 (+) born x = born x (+) born y1 (+) born y2
by ORDINAL7:68;
A291: born y2L (+) born x c= D by A284,A278,ORDINAL1:def 2;
A292: y2L*x = xy2L & y1*x = xy1
by A284,A278,A279,A3,ORDINAL1:def 2;
A293: born y2L (+) born xR c= D by A286,A278,A284,ORDINAL1:10,
ORDINAL1:def 2;
A294: y2L*xR = xRy2L by A3,A287,ORDINAL1:def 2;
A295: y1*xR =xRy1 by A278,A279,A3;
A296: born y2L (+) born y1 in born y1 (+) born y2
by A283,SURREALO:1,ORDINAL7:94;
then A297: born y2L (+) born y1 (+) born x in
born y1 (+) born y2 (+) born x by ORDINAL7:94;
then A298: born y2L (+) born y1 (+) born x in E by A241,A290;
born y2L (+) born y1 (+) born xR in
born y2 (+) born y1 (+) born xR &
born y2 (+) born y1 (+) born xR in
born y2 (+) born y1 (+) born x by A285,SURREALO:1,A296,ORDINAL7:94;
then A299:born y2L (+) born y1 (+) born xR in
born y1 (+) born y2 (+) born x by ORDINAL1:10;
per cases by A244;
suppose A300:y1 < y2L;
born x (+) born y1 (+) born y2L in E by A298,ORDINAL7:68;
then A301: xy2L+xRy1 < xy1+xRy2L
by A277,A159,A280,A300,A293,A278,A279,A291;
xy2 + xRy2L <= (xRy2) + xy2L by Th42,A288;
then A302: xy2 + xRy2L +(xy2L+xRy1) < (xRy2) + xy2L +(xy1+xRy2L)
by A301,Th44;
A303: xy2 + xRy2L +(xy2L+xRy1) = xy2 + (xRy2L +(xy2L+xRy1))
by Th37
.= xy2 + ((xRy2L +xy2L)+xRy1) by Th37
.= xy2+ xRy1 + (xRy2L +xy2L) by Th37;
xRy2 + xy2L +(xy1+xRy2L) = xy2L +(xRy2+(xy1+xRy2L)) by Th37
.= xy2L +((xRy2+xy1)+xRy2L) by Th37
.= xy1+xRy2 +(xRy2L +xy2L) by Th37;
hence xy2+ xRy1 < xy1+xRy2 by A303,A302,Th43;
end;
suppose A304: y1 == y2L;
A305: xy1 == xy2L by A159,A297,A241,A290,A278,A291,A304,A292;
A306: xRy1 == xRy2L
by A293,A295,A294,A278,A304,A299,A159,A241,A290;
A307: xy2 + xRy1 <= xy2 + xRy2L by A306,Th32;
A308: xRy2 + xy2L <= xy1+xRy2 by A305,Th32;
xy2+ xRy1 < xRy2 + xy2L by A289,A307,SURREALO:4;
hence xy2+ xRy1 < xy1+xRy2 by A308,SURREALO:4;
end;
end;
hence thesis by A245;
end;
suppose ex y1R be Surreal st y1R in R_y1 & y1 < y1R <= y2;
then consider y1R be Surreal such that
A309: y1R in R_y1 & y1 < y1R <= y2;
A310: for xL be Surreal st xL in L_x holds P3[D,xL,x,y1,y2]
proof
let xL be Surreal such that A311: xL in L_x;
let xLy2,xy1,xLy1,xy2 be Surreal such that
A312:born xL (+) born y1 c= D & born x (+) born y1 c= D &
born xL (+) born y2 c= D & born x (+) born y2 c= D and
A313: xLy1 = xL*y1 & xLy2=xL*y2 & xy1=x*y1 & xy2=x*y2 and
A314: xL < x & y1 < y2;
A315: xy1 = [comp(L_x,x,y1,L_y1) \/ comp(R_x,x,y1,R_y1),
comp(L_x,x,y1,R_y1) \/ comp(R_x,x,y1,L_y1)] by A313,Th50;
A316: L_xy1 << {xy1} << R_xy1 & xy1 in {xy1}
by SURREALO:11,TARSKI:def 1;
A317: y1R in L_y1 \/ R_y1 by A309,XBOOLE_0:def 3;
A318: born x (+) born y1R in born x (+) born y1
by A317,SURREALO:1,ORDINAL7:94;
reconsider xy1R=x*y1R as Surreal by A2,A312,A318;
A319: xL in L_x \/ R_x by A311,XBOOLE_0:def 3;
then A320: born xL (+) born y1R in born x (+) born y1R
by SURREALO:1,ORDINAL7:94;
then A321: born xL (+) born y1R in born x (+) born y1
by A318,ORDINAL1:10;
A322: born xL (+) born y1R in D by A320,A318,ORDINAL1:10,A312;
reconsider xLy1R=xL*y1R as Surreal by A321,A2,A312;
(xLy1) +' xy1R +' -' xLy1R in comp(L_x,x,y1,R_y1)
by A311,A309,A313,Def14;
then (xLy1) +' xy1R +' -' xLy1R in R_xy1 by A315,XBOOLE_0:def 3;
then A323: xy1 < xLy1 + xy1R - xLy1R by A316;
then A324: xy1 + xLy1R < xLy1 + xy1R by Th42;
A325: born y1 (+) born y2 (+) born x = born x (+) born y1 (+) born y2
by ORDINAL7:68;
A326: born y1R (+) born x c= D by A318,A312,ORDINAL1:def 2;
A327: y1R*x = xy1R & y2*x = xy2 by A318,A312,ORDINAL1:def 2,A313,A3;
A328: born y1R (+) born xL c= D
by A320,A318,ORDINAL1:10,A312,ORDINAL1:def 2;
A329: y1R*xL = xLy1R by A3,A322,ORDINAL1:def 2;
A330: y2*xL =xLy2 by A312,A313,A3;
A331: born y1R (+) born y2 in born y1 (+) born y2
by A317,SURREALO:1,ORDINAL7:94;
then A332: born y1R (+) born y2 (+) born x in
born y1 (+) born y2 (+) born x by ORDINAL7:94;
then A333: born y1R (+) born y2 (+) born x in E by A241,A325;
born y1R (+) born y2 (+) born xL in
born y1 (+) born y2 (+) born xL &
born y1 (+) born y2 (+) born xL in
born y1 (+) born y2 (+) born x by A319,SURREALO:1,A331,ORDINAL7:94;
then A334:born y1R (+) born y2 (+) born xL in
born y1 (+) born y2 (+) born x by ORDINAL1:10;
per cases by A309;
suppose A335:y1R < y2;
born x (+) born y1R (+) born y2 in E by A333,ORDINAL7:68;
then A336: xLy2+xy1R < xLy1R+xy2
by A159,A311,A314,A335,A328,A312,A313,A326;
xy1 + xLy1R <= xLy1 + xy1R by A323,Th42;
then A337: xy1 + xLy1R + (xLy2+xy1R) < xLy1 + xy1R+(xLy1R+xy2)
by A336,Th44;
A338: xy1 + xLy1R + (xLy2+xy1R) = xy1 + (xLy1R + (xLy2+xy1R))
by Th37
.=xy1 + (xLy2+(xLy1R+xy1R)) by Th37
.=xy1 + xLy2+(xLy1R+xy1R) by Th37;
xLy1 + xy1R+(xLy1R+xy2) = xLy1 + (xy1R+(xLy1R+xy2)) by Th37
.= xLy1 + ((xy1R+xLy1R)+xy2) by Th37
.= xLy1 + xy2+(xy1R+xLy1R) by Th37;
hence xLy2+ xy1 < xLy1+xy2 by A337,A338,Th43;
end;
suppose A339: y1R == y2;
A340: xy1R == xy2 by A332,A241,A325,A159,A312,A326,A339,A327;
A341: xLy1R == xLy2
by A328,A330,A329,A312,A339,A334,A159,A241,A325;
A342: xy1 + xLy2 <= xy1 + xLy1R by A341,Th32;
A343:xLy1 + xy1R <= xLy1 + xy2 by A340,Th32;
xy1 + xLy2 < xLy1 + xy1R by A324,A342,SURREALO:4;
hence xLy2+ xy1 < xLy1+xy2 by A343,SURREALO:4;
end;
end;
for xR be Surreal st xR in R_x holds P3[D,x,xR,y1,y2]
proof
let xR be Surreal such that A344: xR in R_x;
let xy2,xRy1,xy1,xRy2 be Surreal such that
A345:born x (+) born y1 c= D & born xR (+) born y1 c= D &
born x (+) born y2 c= D & born xR (+) born y2 c= D and
A346: xy1 = x*y1 & xy2=x*y2 & xRy1=xR*y1 & xRy2=xR*y2 and
A347: x < xR & y1 < y2;
A348: xy1 = [comp(L_x,x,y1,L_y1) \/ comp(R_x,x,y1,R_y1),
comp(L_x,x,y1,R_y1) \/ comp(R_x,x,y1,L_y1)] by A346,Th50;
A349: L_xy1 << {xy1} << R_xy1 & xy1 in {xy1}
by SURREALO:11,TARSKI:def 1;
A350: y1R in L_y1 \/ R_y1 by A309,XBOOLE_0:def 3;
A351: born x (+) born y1R in born x (+) born y1
by SURREALO:1,A350,ORDINAL7:94;
then reconsider xy1R=x*y1R as Surreal by A2,A345;
A352: xR in L_x \/ R_x by A344,XBOOLE_0:def 3;
A353: born xR (+) born y1R in born x (+) born y1R
by A352,SURREALO:1,ORDINAL7:94;
then A354: born xR (+) born y1R in D by A345,A351,ORDINAL1:10;
then reconsider xRy1R=xR*y1R as Surreal by A2;
(xRy1) +' xy1R +' -' xRy1R in comp(R_x,x,y1,R_y1)
by A344,A309,A346,Def14;
then (xRy1) +' xy1R +' -' xRy1R in L_xy1 by A348,XBOOLE_0:def 3;
then A355: xRy1 + xy1R - xRy1R <xy1 by A349;
then A356: xRy1 + xy1R < xy1 + xRy1R by Th41;
A357: born y1 (+) born y2 (+) born x = born x (+) born y1 (+) born y2
by ORDINAL7:68;
A358: born y1R (+) born x c= D by A351,A345,ORDINAL1:def 2;
A359: y1R*x = xy1R & y2*x = xy2
by A345,A346,A3,A351,ORDINAL1:def 2;
A360: born y1R (+) born xR c= D by A353,A345,A351,ORDINAL1:10,
ORDINAL1:def 2;
A361: y1R*xR = xRy1R by A3,A354,ORDINAL1:def 2;
A362: y2*xR =xRy2 by A345,A346,A3;
A363: born y1R (+) born y2 in born y1 (+) born y2
by A350,SURREALO:1,ORDINAL7:94;
then A364: born y1R (+) born y2 (+) born x in
born y1 (+) born y2 (+) born x by ORDINAL7:94;
then A365: born y1R (+) born y2 (+) born x in E by A241,A357;
born y1R (+) born y2 (+) born xR in
born y1 (+) born y2 (+) born xR &
born y1 (+) born y2 (+) born xR in
born y1 (+) born y2 (+) born x by A352,SURREALO:1,A363,ORDINAL7:94;
then
A366: born y1R (+) born y2 (+) born xR in
born y1 (+) born y2 (+) born x by ORDINAL1:10;
per cases by A309;
suppose A367:y1R < y2;
born x (+) born y1R (+) born y2 in E by A365,ORDINAL7:68;
then A368: xy2+xRy1R < xy1R+xRy2
by A344,A159,A347,A367,A360,A345,A346,A358;
xRy1 + xy1R <= xy1 + xRy1R by A355,Th41;
then A369: xRy1 + xy1R +(xy2+xRy1R) < xy1 + xRy1R +(xy1R+xRy2)
by A368,Th44;
A370: xRy1 + xy1R +(xy2+xRy1R) = xRy1 + (xy1R +(xRy1R+xy2))
by Th37
.= xRy1 + ((xy1R +xRy1R)+xy2) by Th37
.=(xy1R + xRy1R)+ (xy2+xRy1) by Th37;
xy1 + xRy1R +(xy1R+xRy2) = xy1 + (xRy1R +(xy1R+xRy2)) by Th37
.= xy1 + ((xRy1R +xy1R)+xRy2) by Th37
.= xy1 +xRy2 +(xRy1R +xy1R) by Th37;
hence xy2+ xRy1 < xy1+xRy2 by A370,A369,Th43;
end;
suppose A371: y1R == y2;
A372: xy1R == xy2 by A159,A364,A241,A357,A345,A358,A371,A359;
A373: xRy1R == xRy2
by A360,A362,A361,A345,A371,A366,A159,A241,A357;
A374: xy1+ xRy1R <= xy1+ xRy2 by A373,Th32;
A375: xy2 + xRy1 <= xy1R + xRy1 by A372,Th32;
xRy1 + xy1R < xy1 + xRy2 by A356,A374,SURREALO:4;
hence xy2+ xRy1 < xy1+xRy2 by A375,SURREALO:4;
end;
end;
hence thesis by A310;
end;
end;
A376:for E be Ordinal holds Q24[E] from ORDINAL1:sch 2(A158);
thus for x1,x2,y be Surreal holds P2[D,x1,x2,y]
proof
let x1,x2,y be Surreal;
born x1 (+) born x2 (+) born y c= born x1 (+) born x2 (+) born y;
hence P2[D,x1,x2,y] by A376;
end;
defpred QEE[Ordinal] means
for x1,x2,y1,y2 be Surreal st born x1 (+) born x2 c= $1
holds P3[D,x1,x2,y1,y2];
A377: for E be Ordinal st for C be Ordinal st C in E holds QEE[C]
holds QEE[E]
proof
let E be Ordinal such that A378:for C be Ordinal st C in E holds QEE[C];
let x1,x2,y1,y2 be Surreal such that
A379: born x1 (+) born x2 c= E;
let x1y2,x2y1,x1y1,x2y2 be Surreal such that
A380: born x1 (+) born y1 c= D & born x2 (+) born y1 c= D &
born x1 (+) born y2 c= D & born x2 (+) born y2 c= D and
A381:x1y1=x1*y1 & x1y2=x1*y2 & x2y1=x2*y1 & x2y2=x2*y2 &
x1 < x2 & y1 < y2;
per cases by A381,SURREALO:13;
suppose ex x1R be Surreal st x1R in R_x1 & x1 < x1R <= x2;
then consider x1R be Surreal such that
A382: x1R in R_x1 & x1 < x1R <= x2;
x1R in L_x1 \/ R_x1 by A382,XBOOLE_0:def 3;
then A383: born x1R (+) born x2 in born x1 (+) born x2 &
born x1R (+) born y1 in born x1 (+) born y1 &
born x1R (+) born y2 in born x1 (+) born y2 by SURREALO:1,ORDINAL7:94;
then reconsider x1Ry1=x1R*y1,
x1Ry2=x1R*y2
as Surreal by A2,A380;
A384: born x1R (+) born y1 c= D & born x1R (+) born y2 c= D
by A383,A380,ORDINAL1:def 2;
per cases by A382;
suppose A385: x1R == x2;
born x1R (+) born x2 (+) born y1 c=born x1R (+) born x2 (+) born y1;
then A386:x1Ry1 == x2y1 by A376,A385,A384,A380,A381;
born x1R (+) born x2 (+) born y2 c=born x1R (+) born x2 (+) born y2;
then A387: x1Ry2 == x2y2 by A376,A385,A384,A380,A381;
born x1 (+) born y1 (+) born y2 c=born x1 (+) born y1 (+) born y2;
then A388: x1y2 + x1Ry1 < x1y1 + x1Ry2 by A376,A382,A380,A381,A384;
A389:x1y1 + x1Ry2 <= x1y1+x2y2 by A387,Th32;
x1y2+ x2y1 <= x1y2+x1Ry1 by A386,Th32;
then x1y2+ x2y1 < x1y1 + x1Ry2 by A388,SURREALO:4;
hence x1y2+x2y1 < x1y1+x2y2 by SURREALO:4,A389;
end;
suppose x1R < x2;
then A390: x1Ry2+x2y1 < x1Ry1+x2y2 by A384,A383,A378,A379,A380,A381;
born x1 (+) born y1 (+) born y2 c= born x1 (+) born y1 (+) born y2;
then x1y2+x1Ry1 <= x1y1 +x1Ry2 by A376,A384,A382,A380,A381;
then A391: x1Ry2+x2y1 +(x1y2+x1Ry1) < x1Ry1+x2y2 +(x1y1 +x1Ry2)
by A390,Th44;
A392: x1Ry2+x2y1 +(x1y2+x1Ry1) = x1Ry2+(x2y1 +(x1y2+x1Ry1)) by Th37
.= x1Ry2+((x2y1 +x1y2)+x1Ry1) by Th37
.= x1Ry2+x1Ry1+ (x1y2+x2y1) by Th37;
x1Ry1+x2y2 +(x1y1 +x1Ry2) = x1Ry1+(x2y2 +(x1y1 +x1Ry2)) by Th37
.= x1Ry1+((x2y2 +x1y1) +x1Ry2) by Th37
.= x1Ry1 +x1Ry2 +(x2y2 +x1y1) by Th37;
hence x1y2+x2y1 < x1y1+x2y2 by A391,A392,Th43;
end;
end;
suppose ex x2L be Surreal st x2L in L_x2 & x1 <= x2L < x2;
then consider x2L be Surreal such that
A393: x2L in L_x2 & x1 <= x2L < x2;
x2L in L_x2 \/ R_x2 by A393,XBOOLE_0:def 3;
then A394: born x2L (+) born x1 in born x1 (+) born x2 &
born x2L (+) born y1 in born x2 (+) born y1 &
born x2L (+) born y2 in born x2 (+) born y2 by SURREALO:1,ORDINAL7:94;
then reconsider x2Ly1=x2L*y1,x2Ly2=x2L*y2 as Surreal by A2,A380;
A395: born x2L (+) born y1 c= D & born x2L (+) born y2 c= D
by A394,A380,ORDINAL1:def 2;
per cases by A393;
suppose A396: x1 == x2L;
born x1 (+) born x2L (+) born y1 c=born x1 (+) born x2L (+) born y1;
then A397:x1y1 == x2Ly1 by A376,A396,A395,A380,A381;
born x1 (+) born x2L (+) born y2 c=born x1 (+) born x2L (+) born y2;
then A398: x1y2 == x2Ly2 by A376,A396,A395,A380,A381;
born x2 (+) born y1 (+) born y2 c=born x2 (+) born y1 (+) born y2;
then A399: x2Ly2 + x2y1 < x2Ly1 + x2y2 by A376,A393,A380,A381,A395;
A400: x2Ly1 + x2y2 <=x1y1+x2y2 by A397,Th32;
x1y2+x2y1 <= x2Ly2 + x2y1 by A398,Th32;
then x1y2+x2y1 < x2Ly1 + x2y2 by A399,SURREALO:4;
hence x1y2+x2y1 < x1y1+x2y2 by A400,SURREALO:4;
end;
suppose x1 < x2L;
then A401: x1y2+x2Ly1 < x1y1+x2Ly2 by A395,A394,A378,A379,A380,A381;
born x2 (+) born y1 (+) born y2 c= born x2 (+) born y1 (+) born y2;
then x2Ly2+x2y1 <= x2Ly1 +x2y2 by A376,A395,A393,A380,A381;
then A402: x2Ly2+x2y1+(x1y2+x2Ly1) < x2Ly1 +x2y2 +(x1y1+x2Ly2)
by A401,Th44;
A403: x2Ly2+x2y1+(x1y2+x2Ly1) = x2Ly2+(x2y1+(x1y2+x2Ly1)) by Th37
.= x2Ly2+((x2y1+x1y2)+x2Ly1) by Th37
.= (x1y2 + x2y1)+(x2Ly2+x2Ly1) by Th37;
x2Ly1 +x2y2 +(x1y1+x2Ly2) = x2Ly1 +(x2y2 +(x1y1+x2Ly2)) by Th37
.= x2Ly1 +((x2y2 +x1y1)+x2Ly2) by Th37
.= (x1y1+x2y2) + (x2Ly2+x2Ly1) by Th37;
hence x1y2+x2y1 < x1y1+x2y2 by A402,A403,Th43;
end;
end;
end;
A404:for E be Ordinal holds QEE[E] from ORDINAL1:sch 2(A377);
let x1,x2,y1,y2 be Surreal;
born x1 (+) born x2 c= born x1 (+) born x2;
hence thesis by A404;
end;
A405:for E be Ordinal holds Q[E] from ORDINAL1:sch 2(A1);
thus for x,y holds x*y is Surreal by A405;
thus for x,y holds x*y = y*x by A405;
thus for x1,x2,y,x1y,x2y be Surreal st x1 == x2 & x1y = x1*y & x2y = x2*y
holds x1y == x2y
proof
let x1,x2,y,x1y,x2y be Surreal such that
A406:x1 == x2 & x1y = x1*y & x2y = x2*y;
set B1 = born x1 (+) born y,B2 = born x2 (+) born y,B=B1\/B2;
B1 c= B & B2 c= B by XBOOLE_1:7;
hence thesis by A406,A405;
end;
let x1,x2,y1,y2,x1y2,x2y1,x1y1,x2y2 be Surreal such that
A407:x1y1=x1*y1 & x1y2=x1*y2 & x2y1=x2*y1 & x2y2=x2*y2 & x1 < x2 & y1 < y2;
set B11 = born x1 (+) born y1,
B12 = born x1 (+) born y2,
B21 = born x2 (+) born y1,
B22 = born x2 (+) born y2,
B1 = B11\/B12, B2 = B21\/B22,B=B1\/B2;
B11 c= B1 & B12 c= B1 & B21 c= B2 & B22 c= B2 &
B1 c= B1\/B2 & B2 c= B1\/B2 by XBOOLE_1:7;
then B11 c= B1\/B2 & B12 c= B1\/B2 & B21 c= B1\/B2 & B22 c= B1\/B2
by XBOOLE_1:1;
hence x1y2+x2y1 < x1y1+x2y2 by A405,A407;
end;
registration
let a,b be Surreal;
cluster a * b -> surreal;
coherence by Th51;
end;
::$N Commutativity of Multiplication for Surreal Number, Conway Ch.1 Th.7(iii)
definition
let a,b be Surreal;
redefine func a * b; :: redefines * (SURREALR, definition)
commutativity by Th51;
end;
registration
let x,y be Surreal;
let X,Y be set;
cluster comp(X,x,y,Y) -> surreal-membered;
coherence
proof
let o;
assume o in comp(X,x,y,Y);
then ex x1,y1 be Surreal st
o = (x1*y) +' (x*y1) +' -' (x1*y1) & x1 in X & y1 in Y by Def14;
hence thesis;
end;
end;
definition
let x,y be Surreal;
let X,Y be set;
redefine func comp(X,x,y,Y) means :Def15: :: redefines comp (SURREALR, definition)
o in it iff ex x1,y1 be Surreal st
o = x1*y + x*y1 - x1*y1 & x1 in X & y1 in Y;
compatibility
proof
let C be set;
thus C =comp(X,x,y,Y) implies
for o holds o in C iff ex x1,y1 be Surreal st
o = (x1*y) + (x*y1) - (x1*y1) & x1 in X & y1 in Y
proof
assume A1:C =comp(X,x,y,Y);
let o;
thus o in C implies ex x1,y1 be Surreal st
o = (x1*y) + (x*y1) - (x1*y1) & x1 in X & y1 in Y
proof
assume o in C;
then consider x1,y1 be Surreal such that
A2:o = (x1*y) +' (x*y1) +' -' (x1*y1) & x1 in X & y1 in Y by A1,Def14;
o = (x1*y) + (x*y1) - (x1*y1) by A2;
hence thesis by A2;
end;
assume ex x1,y1 be Surreal st
o = (x1*y) + (x*y1) - (x1*y1) & x1 in X & y1 in Y;
hence thesis by A1,Def14;
end;
assume A3:for o holds o in C iff ex x1,y1 be Surreal st
o = (x1*y) + (x*y1) - (x1*y1) & x1 in X & y1 in Y;
thus C c= comp(X,x,y,Y)
proof
let o such that A4:o in C;
ex x1,y1 be Surreal st
o = (x1*y) + (x*y1) - (x1*y1) & x1 in X & y1 in Y by A4,A3;
hence thesis by Def14;
end;
let o such that A5:o in comp(X,x,y,Y);
consider x1,y1 be Surreal such that
A6:o = (x1*y) +' (x*y1) +' -' (x1*y1) & x1 in X & y1 in Y by A5,Def14;
o = (x1*y) + (x*y1) - (x1*y1) by A6;
hence thesis by A6,A3;
end;
end;
theorem
comp({z},x,y,{t}) = { z*y + x*t - z*t}
proof
A1:comp({z},x,y,{t}) c= { z*y + x*t - z*t}
proof
let o;
assume o in comp({z},x,y,{t});
then consider x1,y1 be Surreal such that
A2: o = x1*y + x*y1 - x1*y1 & x1 in {z} & y1 in {t} by Def15;
x1=z & y1=t by A2,TARSKI:def 1;
hence thesis by A2,TARSKI:def 1;
end;
z in {z} & t in {t} by TARSKI:def 1;
then z*y + x*t - z*t in comp({z},x,y,{t}) by Def15;
hence thesis by A1, ZFMISC_1:33;
end;
Lm6:for X,Y be set holds comp(X,x,y,Y) c= comp(Y,y,x,X)
proof
let X,Y be set;
let o;
assume o in comp(X,x,y,Y);
then consider x1,y1 be Surreal such that
A1: o = (x1*y) +' (x*y1) +' -' (x1*y1) & x1 in X & y1 in Y by Def14;
o = (x1*y) + (x*y1) + - (x1*y1) by A1
.= (y1*x) + (y*x1) + - (y1*x1);
hence thesis by A1,Def14;
end;
theorem Th53:
for X,Y be set holds comp(X,x,y,Y) = comp(Y,y,x,X)
proof
let X,Y be set;
comp(X,x,y,Y) c= comp(Y,y,x,X) c= comp(X,x,y,Y) by Lm6;
hence thesis by XBOOLE_0:def 10;
end;
::$N Conway Ch.1 Th.8(i)
theorem
for x1,x2,y be Surreal st x1 == x2 holds x1 * y == x2 * y by Th51;
::$N Conway Ch.1 Th.8(iii)
theorem
for x1,x2,y1,y2 be Surreal st x1 < x2 & y1 < y2
holds x1*y2+x2*y1 < x1*y1+x2*y2 by Th51;
::$N Conway Ch.1 Th.7(i)
theorem Th56:
x * 0_No = 0_No
proof
A1: comp(L_x,x,0_No,L_0_No) ={} & comp(R_x,x,0_No,R_0_No) = {} by Th49;
x * 0_No = [comp(L_x,x,0_No,L_0_No) \/ comp(R_x,x,0_No,R_0_No),
comp(L_x,x,0_No,L_0_No) \/ comp(R_x,x,0_No,L_0_No)] by Th50;
hence thesis by A1;
end;
::$N Multiplicative Identity for Surreal Number, Conway Ch.1 Th.7(ii)
theorem Th57:
x * 1_No = x
proof
defpred P[Ordinal] means for x st born x c= $1 holds x*1_No =x;
A1: for D be Ordinal st for C be Ordinal st C in D holds P[C] holds P[D]
proof
let D be Ordinal such that A2: for C be Ordinal st C in D holds P[C];
let x such that A3:born x c= D;
A4: comp(R_x,x,1_No,R_1_No) = {} & comp(L_x,x,1_No,R_1_No) ={} by Th49;
A5: for X be surreal-membered set st X c= L_x \/R_x
holds comp(X,x,1_No,L_1_No)=X
proof
let X be surreal-membered set such that A6:X c= L_x \/R_x;
thus comp(X,x,1_No,L_1_No)c=X
proof
let o such that A7: o in comp(X,x,1_No,L_1_No);
consider x1,y1 be Surreal such that
A8: o = (x1*1_No) +' (x*y1) +' -' (x1*y1) & x1 in X & y1 in L_1_No
by A7,Def14;
A9:y1=0_No by A8,TARSKI:def 1;
A10: born x1 in born x by A8,A6,SURREALO:1;
o = x1*1_No + x*y1 +- 0_No by A8,A9,Th56
.= x1*1_No + 0_No + 0_No by A9,Th56
.= x1*1_No;
hence thesis by A10,A2,A3,A8;
end;
let o such that A11: o in X;
reconsider x1=o as Surreal by A11,SURREAL0:def 16;
A12: born x1 in born x by A11,A6,SURREALO:1;
0_No in L_1_No by TARSKI:def 1;
then
A13: x1*1_No + x*0_No + - x1*0_No in comp(X,x,1_No,L_1_No) by A11,Def14;
x1*1_No + x*0_No + - x1*0_No = x1*1_No + x*0_No +- 0_No by Th56
.= x1*1_No + 0_No + 0_No by Th56
.= x1*1_No;
hence thesis by A12,A2,A3,A13;
end;
A14: comp(L_x,x,1_No,L_1_No)=L_x & comp(R_x,x,1_No,L_1_No)=R_x
by A5,XBOOLE_1:7;
x * 1_No = [comp(L_x,x,1_No,L_1_No) \/ comp(R_x,x,1_No,R_1_No),
comp(L_x,x,1_No,R_1_No) \/ comp(R_x,x,1_No,L_1_No)] by Th50
.=[L_x,R_x] by A4,A14;
hence thesis;
end;
for D be Ordinal holds P[D] from ORDINAL1:sch 2(A1);
hence thesis;
end;
registration
let x;
reduce x*0_No to 0_No;
reducibility by Th56;
reduce x*1_No to x;
reducibility by Th57;
end;
Lm7:
for X,Y,Z be surreal-membered set
st (X = L_x or X = R_x or X={x}) &
(Y = L_y or Y = R_y or Y={y}) &
(Z = L_z or Z = R_z or Z={z}) &
(X <> {x} or Y<>{y} or Z<>{z})
for x1,y1,z1 be Surreal st
x1 in X & y1 in Y & z1 in Z holds
born x1 (+) born y1 (+) born z1 in born x (+) born y (+) born z
proof
set D=born x (+) born y (+) born z;
let X,Y,Z be surreal-membered set such that
A1: (X = L_x or X = R_x or X={x}) &
(Y = L_y or Y = R_y or Y={y}) &
(Z = L_z or Z = R_z or Z={z}) &
(X <> {x} or Y<>{y} or Z<>{z});
let x1,y1,z1 be Surreal such that
A2: x1 in X & y1 in Y & z1 in Z;
per cases;
suppose Z<>{z};
then A3: z1 in L_z \/ R_z by A1,A2,XBOOLE_0:def 3;
x1 in L_x\/R_x or x1=x by A2,A1,TARSKI:def 1,XBOOLE_0:def 3;
then A4: born x1 c= born x by SURREALO:1,ORDINAL1:def 2;
y1 in L_y\/R_y or y1=y by A2,A1,TARSKI:def 1,XBOOLE_0:def 3;
then born y1 c= born y by SURREALO:1,ORDINAL1:def 2;
then born x1 (+) born y1 c= born x1 (+) born y c= born x (+) born y
by A4,ORDINAL7:95;
then born x1 (+) born y1 c= born x (+) born y by XBOOLE_1:1;
then born x1 (+) born y1 (+) born z1 c= born x (+) born y (+) born z1
in born x (+) born y (+) born z
by A3,SURREALO:1,ORDINAL7:94,95;
hence thesis by ORDINAL1:12;
end;
suppose A5:Z={z};
then (x1 in L_x\/R_x & y1 in L_y\/R_y) or
(x1 in L_x\/R_x & y1 =y) or
(x1 = x & y1 in L_y\/R_y) by A1,A2,XBOOLE_0:def 3,TARSKI:def 1;
then born x1 (+) born y1 in born x (+) born y or
born x1 (+) born y1 in born x1 (+) born y in born x (+) born y
by ORDINAL7:94,SURREALO:1;
then born x1 (+) born y1 in born x (+) born y
by ORDINAL1:10;
then born x1 (+) born y1 (+) born z1 in born x (+) born y (+) born z1
by ORDINAL7:94;
hence thesis by A2,A5,TARSKI:def 1;
end;
end;
Lm8:for x,y be Surreal holds -(x*y) = (-x) * y
proof
defpred P[Ordinal] means
for x,y be Surreal st born x (+) born y c= $1 holds -(x*y) = (-x) * y;
A1: for D be Ordinal st for C be Ordinal st C in D holds P[C] holds P[D]
proof
let D be Ordinal such that A2: for C be Ordinal st C in D holds P[C];
let x,y be Surreal such that A3: born x (+) born y c= D;
set mx=-x;
A4: mx = [--R_x, --L_x] by Th7;
A5: for X,Y be surreal-membered set st
X c= L_x \/ R_x & Y c= L_y\/R_y holds comp(--X,-x,y,Y) = -- comp(X,x,y,Y)
proof
let X,Y be surreal-membered set such that
A6:X c= L_x \/ R_x & Y c= L_y \/ R_y;
thus comp(--X,-x,y,Y) c= -- comp(X,x,y,Y)
proof
let o;
assume o in comp(--X,-x,y,Y);
then consider x1,y1 be Surreal such that
A7:o = x1*y + mx*y1 - x1*y1 & x1 in --X & y1 in Y by Def15;
consider z be Surreal such that
A8: z in X & x1= -z by A7,Def4;
A9: (z*y) + (x*y1) -(z*y1) in comp(X,x,y,Y) by A7,A8,Def15;
born z (+) born y in born x (+) born y
by A8,A6,SURREALO:1,ORDINAL7:94;
then A10: -(z*y) = x1 * y by A8,A2,A3;
A11:born x (+) born y1 in born x (+) born y
by A7,A6,SURREALO:1,ORDINAL7:94;
then A12: -(x*y1) = mx * y1 by A2,A3;
born z (+) born y1 in born x (+) born y1
by A8,A6,SURREALO:1,ORDINAL7:94;
then o = (-(z*y)) + (-(x*y1)) + - (-(z*y1))
by A7,A10,A12,A8,A2,A3,A11,ORDINAL1:10
.= (- ((z*y) + (x*y1))) + - (-(z*y1)) by Th40
.= - ((z*y) + (x*y1) + -(z*y1)) by Th40;
hence thesis by A9,Def4;
end;
let o;
assume o in -- comp(X,x,y,Y);
then consider oo be Surreal such that
A13: oo in comp(X,x,y,Y) & o= -oo by Def4;
consider z,y1 be Surreal such that
A14:oo = z*y + x*y1 - z*y1 & z in X & y1 in Y by A13,Def15;
set x1 = -z;
A15: x1 in --X by A14,Def4;
born z (+) born y in born x (+) born y by A14,A6,SURREALO:1,ORDINAL7:94;
then A16: -(z*y) = x1 * y by A2,A3;
A17:born x (+) born y1 in born x (+) born y
by A14,A6,SURREALO:1,ORDINAL7:94;
then A18: -(x*y1) = mx * y1 by A2,A3;
born z (+) born y1 in born x (+) born y1
by A14,A6,SURREALO:1,ORDINAL7:94;
then x1 * y + mx * y1 - x1 * y1
= (-(z*y)) + (-(x*y1)) + - (-(z*y1)) by A16,A18,A2,A3,A17,ORDINAL1:10
.= (- ((z*y) + (x*y1))) - (-(z*y1)) by Th40
.= o by A13,A14,Th40;
hence thesis by A15,A14,Def15;
end;
A19: L_x c= L_x \/ R_x & L_y c= L_y \/ R_y &
R_x c= L_x \/ R_x & R_y c= L_y \/ R_y by XBOOLE_1:7;
A20: comp(L_mx,mx,y,L_y) = -- comp(R_x,x,y,L_y) &
comp(L_mx,mx,y,R_y) = -- comp(R_x,x,y,R_y) by A4,A5,A19;
A21: comp(R_mx,mx,y,L_y) = -- comp(L_x,x,y,L_y) &
comp(R_mx,mx,y,R_y) = -- comp(L_x,x,y,R_y) by A4,A5,A19;
A22: x * y = [comp(L_x,x,y,L_y) \/ comp(R_x,x,y,R_y),
comp(L_x,x,y,R_y) \/ comp(R_x,x,y,L_y)] by Th50;
mx * y =
[-- comp(R_x,x,y,L_y) \/ -- comp(L_x,x,y,R_y),
--comp(R_x,x,y,R_y) \/ --comp(L_x,x,y,L_y)] by A20,A21,Th50
.= [-- (comp(R_x,x,y,L_y) \/ comp(L_x,x,y,R_y)),
--comp(R_x,x,y,R_y) \/ --comp(L_x,x,y,L_y)] by Th20
.= [-- R_(x*y), -- L_(x*y)] by A22,Th20
.= - (x*y) by Th7;
hence thesis;
end;
for D be Ordinal holds P[D] from ORDINAL1:sch 2(A1);
hence thesis;
end;
::$N Conway Ch.1 Th.7(in)
theorem Th58:
x * (-y) = -(x*y) & (-x) * y = -(x*y) & (-x) * (-y) = x*y
proof
(-x) * (-y) = - ( (-x)*y) by Lm8;
then (-x) * (-y) = - -(x*y) by Lm8;
hence thesis by Lm8;
end;
theorem Th59:
for X,Y1,Y2 be set st Y1 c= Y2 holds comp(X,x,y,Y1) c= comp(X,x,y,Y2)
proof
let X,Y1,Y2 be set such that A1: Y1 c= Y2;
let o;
assume o in comp(X,x,y,Y1);
then ex x1,y1 be Surreal st
o = (x1*y) +' (x*y1) +' -' (x1*y1) & x1 in X & y1 in Y1 by Def14;
hence thesis by A1,Def14;
end;
theorem Th60:
for X,Y1,Y2 be set holds
comp(X,x,y,Y1\/Y2) = comp(X,x,y,Y1)\/comp(X,x,y,Y2)
proof
let X,Y1,Y2 be set;
thus comp(X,x,y,Y1\/Y2) c= comp(X,x,y,Y1)\/comp(X,x,y,Y2)
proof
let o;
assume o in comp(X,x,y,Y1\/Y2);
then consider x1,y1 be Surreal such that
A1: o = (x1*y) +' (x*y1) +' -' (x1*y1) & x1 in X & y1 in Y1\/Y2 by Def14;
y1 in Y1 or y1 in Y2 by A1,XBOOLE_0:def 3;
then o in comp(X,x,y,Y1) or o in comp(X,x,y,Y2) by A1,Def14;
hence thesis by XBOOLE_0:def 3;
end;
comp(X,x,y,Y1) c= comp(X,x,y,Y1\/Y2) &
comp(X,x,y,Y2) c= comp(X,x,y,Y1\/Y2) by XBOOLE_1:7,Th59;
hence thesis by XBOOLE_1:8;
end;
theorem Th61:
for X,Y be set st
for x st x in X ex y st y in Y & x==y holds X <=_ Y
proof
let X,Y be set such that
A1:for x st x in X ex y st y in Y & x==y;
let x;assume x in X;
then ex y st y in Y & x==y by A1;
hence thesis;
end;
theorem
for X1,X2 be set st X1 <=_ X2 holds -- X1 <=_ -- X2
proof
let X1,X2 be set such that A1:X1 <=_ X2;
let x such that A2: x in -- X1;
consider y such that
A3:y in X1 & x = -y by A2,Def4;
consider y2,y3 be Surreal such that
A4:y2 in X2 & y3 in X2 & y2 <= y <= y3 by A3,A1;
A5:-y2 in --X2 & -y3 in --X2 by A4,Def4;
-y3 <= -y <= -y2 by Th10,A4;
hence thesis by A5,A3;
end;
theorem
for X1,X2 be set holds -- (X1++X2) = (--X1) ++ (--X2)
proof
let X1,X2 be set;
thus -- (X1++X2) c= (--X1) ++ (--X2)
proof
let a be object;
assume a in -- (X1++X2);
then consider xy be Surreal such that
A1: xy in X1++X2 & a=-xy by Def4;
consider x,y such that
A2: x in X1 & y in X2 & xy = x+y by A1,Def8;
-x in --X1 & -y in --X2 by A2,Def4;
then -(x+y) = -x + -y in (--X1) ++ (--X2) by Th40,Def8;
hence thesis by A1,A2;
end;
let a be object;
assume a in (--X1) ++ (--X2);
then consider x1,y1 be Surreal such that
A3: x1 in --X1 & y1 in --X2 & a = x1 + y1 by Def8;
consider x such that
A4: x in X1 & x1 = -x by A3,Def4;
consider y such that
A5: y in X2 & y1 = -y by A3,Def4;
x+y in X1++X2 by A4,A5,Def8;
then -x + -y = -(x+y) in -- (X1++X2) by Th40,Def4;
hence thesis by A3,A4,A5;
end;
theorem
for X be surreal-membered set holds X ++ {0_No} = X
proof
let X be surreal-membered set;
thus X ++ {0_No} c= X
proof
let xy be object such that A1:xy in X ++ {0_No};
consider x,y such that
A2: x in X & y in {0_No} & xy= x+y by A1,Def8;
y=0_No by A2,TARSKI:def 1;
hence thesis by A2;
end;
let xy be object such that A3:xy in X;
reconsider x=xy as Surreal by A3,SURREAL0:def 16;
x+0_No = x &0_No in {0_No} by TARSKI:def 1;
hence thesis by A3,Def8;
end;
theorem Th65:
x==y implies -x== -y by Th10;
theorem
for x1,x2,y1,y2 be Surreal st
x1==x2 & y1==y2 holds x1+y1 == x2+y2 by Th43;
::$N Distributivity of Multiplication Over Addition for Surreal Numbers, Conway Ch.1 Th.7(v)
theorem Th67:
x*(y+z) == x*y + x*z
proof
defpred P[Ordinal] means
for x,y,z be Surreal st
born x (+) born y (+) born z c= $1 holds x*(y+z) == x*y + x*z;
A1: for D be Ordinal st for C be Ordinal st C in D holds P[C] holds P[D]
proof
let D be Ordinal such that A2: for C be Ordinal st C in D holds P[C];
let x,y,z such that A3:born x (+) born y (+) born z c= D;
set xy=x*y,xz=x*z, yz=y+z;
A4: xy = [comp(L_x,x,y,L_y) \/ comp(R_x,x,y,R_y),
comp(L_x,x,y,R_y) \/ comp(R_x,x,y,L_y)] by Th50;
A5: xz = [comp(L_x,x,z,L_z) \/ comp(R_x,x,z,R_z),
comp(L_x,x,z,R_z) \/ comp(R_x,x,z,L_z)] by Th50;
A6: yz = [(L_y ++ {z})\/({y} ++ L_z), (R_y ++ {z}) \/({y} ++ R_z)]by Th28;
A7:x*yz = [comp(L_x,x,yz,L_yz) \/ comp(R_x,x,yz,R_yz),
comp(L_x,x,yz,R_yz) \/ comp(R_x,x,yz,L_yz)] by Th50;
A8: xy+xz =
[(L_xy ++ {xz})\/({xy} ++ L_xz),(R_xy ++ {xz}) \/({xy} ++ R_xz)] by Th28;
A9:comp(L_x,x,yz,L_yz) = comp(L_x,x,yz,L_y++{z})\/comp(L_x,x,yz,{y}++L_z)
by A6,Th60;
A10:comp(R_x,x,yz,L_yz) = comp(R_x,x,yz,L_y++{z})\/
comp(R_x,x,yz,{y} ++ L_z) by A6,Th60;
A11: comp(R_x,x,yz,R_yz) = comp(R_x,x,yz,R_y ++ {z})
\/comp(R_x,x,yz,{y} ++ R_z) by A6,Th60;
A12: comp(L_x,x,yz,R_yz) = comp(L_x,x,yz,R_y ++ {z})
\/comp(L_x,x,yz,{y} ++ R_z) by A6,Th60;
A13:for X,Z be surreal-membered set st (X=L_x or X=R_x) & (Z=L_z or Z=R_z)
holds comp(X,x,z,Z) ++{xy} <==> comp(X,x,yz,Z ++ {y})
proof
let X,Z be surreal-membered set such that
A14: (X=L_x or X=R_x) &(Z=L_z or Z=R_z);
for o be Surreal st o in comp(X,x,z,Z) ++{xy} ex b be Surreal st
b in comp(X,x,yz,Z ++ {y}) & o==b
proof
let o be Surreal;
assume o in comp(X,x,z,Z) ++{xy};
then consider xz1,xy1 be Surreal such that
A15: xz1 in comp(X,x,z,Z) & xy1 in {xy} & o = xz1 + xy1 by Def8;
consider x1,z1 be Surreal such that
A16:xz1 = x1 * z + x*z1 - x1*z1 & x1 in X & z1 in Z by A15,Def15;
A17:R_x<> {x} <> L_x & R_z<>{z} <> L_z by SURREALO:2;
A18:x in {x} & y in {y} & z in {z} by TARSKI:def 1;
born x1 (+) born y (+) born z in born x (+) born y (+) born z
by A14,A16,A18,A17,Lm7;
then A19: x1 * (y+z) == x1* y + x1*z by A2,A3;
set yz1=y+z1;
born x (+) born y (+) born z1 in born x (+) born y (+) born z
by A14,A18,A17,Lm7,A16;
then A20: x*(y+z1) == x*y + x*z1 by A2,A3;
A21: born x1 (+) born y (+) born z1 in born x (+) born y (+) born z
by A14,A18,A17,Lm7,A16;
A22: - (x1 * z1 + x1* y) = (- (x1 * z1)) + (- (x1* y)) by Th40;
yz1 in {y}++Z by A18,A16,Def8;
then A23:x1 * yz + x*yz1 - x1*yz1 in comp(X,x,yz,{y}++Z)
by A16,Def15;
x1*z + (x*y + x*z1) = x1*z + x*z1 + x*y by Th37;
then A24: (x1*z + (x*y + x*z1)) + (- (x1 * z1))
= x*y + (x1*z + x*z1 + - (x1 * z1)) by Th37;
A25:x1* y - (x1* y)==0_No by Th39;
A26: - x1*yz1 == ((- (x1 * z1)) + (- (x1* y))) by A22,A21,A2,A3,Th65;
x1 * yz + x*yz1 == (x1* y + x1*z) + (x*y + x*z1) by A19,A20,Th43;
then
A27: x1 * yz + x*yz1 + - x1*yz1 == (x1* y + x1*z) + (x*y + x*z1)
+ ((- (x1 * z1)) + (- (x1* y))) by A26,Th43;
(x1* y + x1*z) + (x*y + x*z1) + ((- (x1 * z1)) + (- (x1* y)))
= (x1* y + (x1*z + (x*y + x*z1))) + ((- (x1 * z1)) + (- (x1* y)))
by Th37
.= ((x1* y + (x1*z + (x*y + x*z1))) + (- (x1 * z1))) + - (x1* y)
by Th37
.= x1* y + ((x1*z + (x*y + x*z1)) + (- (x1 * z1))) + - (x1* y) by Th37
.= x1* y + - (x1* y) + ( (x1*z + x*z1 + - (x1 * z1)) +x*y )
by A24,Th37;
then x1 * yz + x*yz1 + - x1*yz1 == x1* y + - (x1* y) + o == 0_No+o = o
by A27,A16,A15,TARSKI:def 1,A25,Th43;
hence thesis by A23,SURREALO:10;
end;
hence comp(X,x,z,Z) ++{xy} <=_ comp(X,x,yz,Z ++ {y}) by Th61;
for o be Surreal st o in comp(X,x,yz,Z ++ {y}) ex b be Surreal st
b in comp(X,x,z,Z) ++{xy} & o==b
proof
let o be Surreal;
assume o in comp(X,x,yz,Z ++ {y});
then consider x1,yz1 be Surreal such that
A28: o = x1 * yz + x*yz1 - x1*yz1 & x1 in X & yz1 in Z ++ {y}
by Def15;
consider z1,y1 be Surreal such that
A29: z1 in Z & y1 in {y} & yz1 = z1 + y1 by A28,Def8;
A30: y1=y by A29,TARSKI:def 1;
A31: R_x<> {x} <> L_x & R_z<>{z} <> L_z by SURREALO:2;
A32: x in {x} & y in {y} & z in {z} by TARSKI:def 1;
born x1 (+) born y (+) born z in born x (+) born y (+) born z
by A14,A32,A31,A28,Lm7;
then A33: x1 * (y+z) == x1* y + x1*z by A2,A3;
born x (+) born y1 (+) born z1 in born x (+) born y (+) born z
by A14,A32,A31,Lm7,A29;
then A34: x*yz1 == x*y + x*z1 by A2,A3,A29,A30;
born x1 (+) born y1 (+) born z1 in born x (+) born y (+) born z
by A14,A31,A28,Lm7,A29;
then A35: x1*yz1 == x1*z1 + x1*y by A2,A3,A29,A30;
A36: x1* z + x*z1 - (x1 * z1) in comp(X,x,z,Z) by A28,A29,Def15;
xy in {xy} by TARSKI:def 1;
then A37: (x1* z + x*z1 + - (x1 * z1))+xy in comp(X,x,z,Z) ++{xy}
by A36,Def8;
x1* y - (x1* y) ==0_No by Th39;
then A38: x1* y + - (x1* y) + ( (x1*z + x*z1 + - (x1 * z1)) +x*y )
== 0_No + ( (x1*z + x*z1 + - (x1 * z1)) +x*y ) by Th43;
x1*z + (x*y + x*z1) = x1*z + x*z1 + x*y by Th37;
then A39: (x1*z + (x*y + x*z1)) + (- (x1 * z1))
= x*y + (x1*z + x*z1 + - (x1 * z1)) by Th37;
- x1*yz1 == - (x1*z1 + x1*y) by A35,Th10;
then A40:- x1*yz1 == (- (x1 * z1)) + (- (x1* y)) by Th40;
x1 * yz + x*yz1 == (x1* y + x1*z) + (x*y + x*z1)
by A33,A34,Th43;
then A41:o == (x1* y + x1*z) + (x*y + x*z1) +
((- (x1 * z1)) + (- (x1* y))) by A40,A28,Th43;
(x1* y + x1*z) + (x*y + x*z1) + ((- (x1 * z1)) + (- (x1* y)))
= (x1* y + (x1*z + (x*y + x*z1))) + ((- (x1 * z1)) + (- (x1* y)))
by Th37
.= ((x1* y + (x1*z + (x*y + x*z1))) + (- (x1 * z1))) + - (x1* y)
by Th37
.= x1* y + ((x1*z + (x*y + x*z1)) + (- (x1 * z1))) + - (x1* y) by Th37
.= x1* y + -(x1* y) + ( (x1*z + x*z1 + - (x1 * z1)) +x*y)by A39,Th37;
then o == 0_No + ((x1*z + x*z1 + - (x1 * z1)) +x*y)
by A41,A38,SURREALO:4;
hence thesis by A37;
end;
hence thesis by Th61;
end;
A42:for X,Y be surreal-membered set st (X=L_x or X=R_x) & (Y=L_y or Y=R_y)
holds comp(X,x,y,Y) ++{xz} <==> comp(X,x,yz,Y ++ {z})
proof
let X,Y be surreal-membered set such that
A43: (X=L_x or X=R_x) & (Y=L_y or Y=R_y);
for o be Surreal st o in comp(X,x,y,Y) ++{xz} ex b be Surreal st
b in comp(X,x,yz,Y ++ {z}) & o==b
proof
let o be Surreal;
assume o in comp(X,x,y,Y) ++{xz};
then consider xy1,xz1 be Surreal such that
A44: xy1 in comp(X,x,y,Y) & xz1 in {xz} & o = xy1 + xz1 by Def8;
consider x1,y1 be Surreal such that
A45:xy1 = x1 * y + x*y1 - x1*y1 & x1 in X & y1 in Y
by A44,Def15;
A46: xz1=xz by A44,TARSKI:def 1;
A47: (x1* y + x1*z) + (x*y1 + x*z)=x1* y + (x1*z + (x*z + x*y1))
by Th37
.=x1* y + ((x1*z + x*z) + x*y1) by Th37
.=(x1* y + x*y1) +(x1*z + x*z) by Th37;
A48: R_x<> {x} <> L_x & R_y<>{y} <> L_y by SURREALO:2;
A49: x in {x} & y in {y} & z in {z} by TARSKI:def 1;
born x1 (+) born y (+) born z in born x (+) born y (+) born z
by A43,A45,A49,A48,Lm7;
then A50: x1 * (y+z) == x1* y + x1*z by A2,A3;
set yz1=y1+z;
born x (+) born y1 (+) born z in born x (+) born y (+) born z
by A43,A49,A48,Lm7,A45;
then A51: x*(y1+z) == x*y1 + x*z by A2,A3;
A52: born x1 (+) born y1 (+) born z in born x (+) born y (+) born z
by A43,A49,A48,Lm7,A45;
x1*z - (x1* z) ==0_No by Th39;
then A53: x*z+(x1*z + - (x1* z)) == x*z+0_No =x *z by Th43;
( x*z+x1*z) + (- (x1* z)) == x*z by A53,Th37;
then A54:( x*z+x1*z) + (- (x1* z)) +(- (x1 * y1)) == x*z +- (x1 * y1)
by Th43;
A55: (x1* y + x*y1) + (x*z +- (x1 * y1))
= x1* y + (x*y1 + (- (x1 * y1)+x*z)) by Th37
.= x1* y + ((x*y1 + - (x1 * y1))+x*z) by Th37
.= x1* y + (x*y1 + - (x1 * y1))+x*z by Th37
.= (x1* y + x*y1 + - (x1 * y1))+x*z by Th37;
A56: - (x1 * y1 + x1* z) = (- (x1 * y1)) + (- (x1* z)) by Th40;
yz1 in Y++{z} by A49,A45,Def8;
then A57:x1 * yz + x*yz1 -x1*yz1 in comp(X,x,yz,Y++{z}) by A45,Def15;
A58: - x1*yz1 == (- (x1* z)) +(- (x1 * y1))
by A52,A2,A3,Th65,A56;
x1 * yz + x*yz1 == (x1* y + x*y1) + ( x*z+x1*z) by A47,A50,A51,Th43;
then x1 * yz + x*yz1 + - x1*yz1 ==
(x1* y + x*y1) + ( x*z+x1*z)+((- (x1* z)) +(- (x1 * y1))) by A58,Th43;
then x1 * yz + x*yz1 + - x1*yz1 ==
(x1* y + x*y1) + (( x*z+x1*z) + ( (- (x1* z)) +(- (x1 * y1))))by Th37;
then A59:x1 * yz + x*yz1 + - x1*yz1 ==
(x1* y + x*y1) + ((( x*z+x1*z) + (- (x1* z))) +(- (x1 * y1)))by Th37;
(x1* y + x*y1) + ((( x*z+x1*z) + (- (x1* z))) +(- (x1 * y1))) ==
(x1* y + x*y1) + (x*z +- (x1 * y1)) by A54,Th43;
hence thesis by A57,A44,A46,A55,A45,A59,SURREALO:10;
end;
hence comp(X,x,y,Y) ++{xz} <=_ comp(X,x,yz,Y ++ {z}) by Th61;
for o be Surreal st o in comp(X,x,yz,Y ++ {z})
ex b be Surreal st b in comp(X,x,y,Y) ++{xz} & o==b
proof
let o be Surreal;
assume o in comp(X,x,yz,Y ++ {z});
then consider x1,yz1 be Surreal such that
A60:o = x1 * yz + x*yz1 - x1*yz1 & x1 in X & yz1 in Y ++ {z}
by Def15;
consider y1,z1 be Surreal such that
A61: y1 in Y & z1 in {z} & yz1 = y1 + z1 by A60,Def8;
A62: z1=z by A61,TARSKI:def 1;
A63: (x1* y + x1*z) + (x*y1 + x*z)
=x1* y + (x1*z + (x*z + x*y1)) by Th37
.=x1* y + ((x1*z + x*z) + x*y1) by Th37
.=(x1* y + x*y1) +(x1*z + x*z) by Th37;
A64: R_x<> {x} <> L_x & R_y<>{y} <> L_y by SURREALO:2;
A65: x in {x} & y in {y} & z in {z} by TARSKI:def 1;
born x1 (+) born y (+) born z in born x (+) born y (+) born z
by A43,A65,A64,A60,Lm7;
then A66: x1 * (y+z) == x1* y + x1*z by A2,A3;
born x (+) born y1 (+) born z1 in born x (+) born y (+) born z
by A43,A65,A64,Lm7,A61;
then A67: x*yz1 == x*y1 + x*z by A2,A3,A61,A62;
A68: born x1 (+) born y1 (+) born z1 in born x (+) born y (+) born z
by A43,A64,A60,Lm7,A61;
A69: x1*z - (x1* z) ==0_No by Th39;
A70: x1* y + x*y1 - (x1 * y1) in comp(X,x,y,Y) by A60,A61,Def15;
xz in {xz} by TARSKI:def 1;
then A71: (x1* y + x*y1 + - (x1 * y1))+xz in comp(X,x,y,Y) ++{xz}
by A70,Def8;
A72: (x1* y + x*y1) + (x*z +- (x1 * y1))
= x1* y + (x*y1 + (- (x1 * y1)+x*z)) by Th37
.= x1* y + ((x*y1 + - (x1 * y1))+x*z) by Th37
.= x1* y + (x*y1 + - (x1 * y1))+x*z by Th37
.= (x1* y + x*y1 + - (x1 * y1))+x*z by Th37;
A73: - (x1 * y1 + x1* z) = (- (x1 * y1)) + (- (x1* z)) by Th40;
A74:- x1*yz1 == (- (x1* z)) +(- (x1 * y1))
by A73,A68,A2,A3,A61,A62,Th65;
A75: x1 * yz + x*yz1 == (x1* y + x*y1) +(x*z+x1*z)
by A66,A67,A63,Th43;
A76: o == (x1* y + x*y1) +(x*z+x1*z) +((- (x1* z)) +(- (x1 * y1)))
by A60,Th43,A74,A75;
A77: (x1* y + x*y1) +(x*z+x1*z) +((- (x1* z)) +(- (x1 * y1)))
= (x1* y + x*y1) + (( x*z+x1*z) + ( (- (x1* y1)) +(- (x1 * z))))
by Th37
.=(x1* y + x*y1) + ( (( x*z+x1*z) + - (x1* y1)) +(- (x1 * z)))
by Th37
.=(x1* y + x*y1) + ( (( x*z+ - (x1* y1) +x1*z )) +(- (x1 * z)))
by Th37
.=(x1* y + x*y1) + ( ( x*z+ - (x1* y1)) +(x1*z +(- (x1 * z))))
by Th37
.= ((x1* y + x*y1 + - (x1 * y1))+x*z)+(( (x1* z)) +(- (x1 * z)))
by A72,Th37;
((x1* y + x*y1 + - (x1 * y1))+x*z)+(( (x1* z)) +(- (x1 * z))) ==
((x1* y + x*y1 + - (x1 * y1))+x*z)+0_No by A69,Th43;
hence thesis by A76,SURREALO:10,A71,A77;
end;
hence thesis by Th61;
end;
A78: comp(L_x,x,yz,L_y ++ {z}) <==> comp(L_x,x,y,L_y) ++{xz} by A42;
comp(L_x,x,yz,{y} ++ L_z) <==> comp(L_x,x,z,L_z) ++{xy} by A13;
then A79: comp(L_x,x,yz,L_yz) <==> (comp(L_x,x,y,L_y) ++{xz})
\/(comp(L_x,x,z,L_z) ++{xy}) by A78,SURREALO:31,A9;
A80:comp(R_x,x,yz,R_y ++ {z}) <==> comp(R_x,x,y,R_y) ++{xz} by A42;
comp(R_x,x,yz,{y} ++ R_z) <==> comp(R_x,x,z,R_z) ++{xy} by A13;
then A81: comp(R_x,x,yz,R_yz) <==> (comp(R_x,x,y,R_y) ++{xz})
\/(comp(R_x,x,z,R_z) ++{xy}) by A80,SURREALO:31,A11;
((comp(L_x,x,y,L_y) ++{xz}) \/(comp(L_x,x,z,L_z) ++{xy})) \/
((comp(R_x,x,y,R_y) ++{xz})\/(comp(R_x,x,z,R_z) ++{xy})) =
(((comp(L_x,x,y,L_y) ++{xz}) \/(comp(L_x,x,z,L_z) ++{xy})) \/
(comp(R_x,x,z,R_z) ++{xy}))\/ (comp(R_x,x,y,R_y) ++{xz}) by XBOOLE_1:4
.= ( (comp(L_x,x,y,L_y) ++{xz}) \/((comp(L_x,x,z,L_z) ++{xy})
\/ (comp(R_x,x,z,R_z) ++{xy})))
\/ (comp(R_x,x,y,R_y) ++{xz}) by XBOOLE_1:4
.= ((comp(L_x,x,y,L_y) ++{xz}) \/ (comp(R_x,x,y,R_y) ++{xz})) \/
((comp(L_x,x,z,L_z) ++{xy}) \/ (comp(R_x,x,z,R_z) ++{xy})) by XBOOLE_1:4
.= ( ( comp(L_x,x,y,L_y) \/ comp(R_x,x,y,R_y)) ++{xz}) \/
((comp(L_x,x,z,L_z) ++{xy}) \/ (comp(R_x,x,z,R_z) ++{xy})) by Th33
.= (xy+xz)`1 by A8,Th33,A5,A4;
then A82: L_(x*yz) <==> L_(xy+xz) by A81,A7,A79,SURREALO:31;
A83: comp(L_x,x,yz,R_y ++ {z}) <==> comp(L_x,x,y,R_y) ++{xz} by A42;
comp(L_x,x,yz,{y} ++ R_z) <==> comp(L_x,x,z,R_z) ++{xy} by A13;
then A84: comp(L_x,x,yz,R_yz) <==> (comp(L_x,x,y,R_y) ++{xz})
\/(comp(L_x,x,z,R_z) ++{xy}) by A83,SURREALO:31,A12;
A85:comp(R_x,x,yz,L_y ++ {z}) <==> comp(R_x,x,y,L_y) ++{xz} by A42;
comp(R_x,x,yz,{y} ++ L_z) <==> comp(R_x,x,z,L_z) ++{xy} by A13;
then A86:comp(R_x,x,yz,L_yz) <==> (comp(R_x,x,y,L_y) ++{xz})
\/(comp(R_x,x,z,L_z) ++{xy})
by A85,SURREALO:31,A10;
A87: R_(x*yz)<==>((comp(L_x,x,y,R_y) ++{xz})\/(comp(L_x,x,z,R_z) ++{xy}))\/
((comp(R_x,x,y,L_y) ++{xz})\/(comp(R_x,x,z,L_z) ++{xy}))
by A7,A84,A86,SURREALO:31;
((comp(L_x,x,y,R_y) ++{xz})\/(comp(L_x,x,z,R_z) ++{xy}))\/
((comp(R_x,x,y,L_y) ++{xz})\/(comp(R_x,x,z,L_z) ++{xy}))=
(((comp(L_x,x,y,R_y) ++{xz})\/(comp(L_x,x,z,R_z) ++{xy}))\/
(comp(R_x,x,z,L_z) ++{xy})) \/ (comp(R_x,x,y,L_y) ++{xz}) by XBOOLE_1:4
.=((comp(L_x,x,y,R_y) ++{xz}) \/((comp(L_x,x,z,R_z) ++{xy})\/
(comp(R_x,x,z,L_z) ++{xy}))) \/ (comp(R_x,x,y,L_y) ++{xz}) by XBOOLE_1:4
.= ( (comp(L_x,x,y,R_y) ++{xz}) \/(comp(R_x,x,y,L_y) ++{xz}))
\/ ((comp(L_x,x,z,R_z) ++{xy})\/(comp(R_x,x,z,L_z) ++{xy})) by XBOOLE_1:4
.=((comp(L_x,x,y,R_y) \/comp(R_x,x,y,L_y)) ++{xz})
\/ ((comp(L_x,x,z,R_z) ++{xy})\/(comp(R_x,x,z,L_z) ++{xy})) by Th33
.= (xy+xz)`2 by Th33,A8,A4,A5;
hence thesis by A87,A7,A8,A82,SURREALO:29;
end;
for D be Ordinal holds P[D] from ORDINAL1:sch 2(A1);
then P[born x (+) born y (+) born z];
hence thesis;
end;
theorem Th68:
for X1,X2,Y be set holds
comp(X1\/X2,x,y,Y) = comp(X1,x,y,Y)\/comp(X2,x,y,Y)
proof
let X1,X2,Y be set;
comp(X1\/X2,x,y,Y) = comp(Y,y,x,X1\/X2) by Th53
.= comp(Y,y,x,X1) \/comp(Y,y,x,X2) by Th60
.= comp(X1,x,y,Y) \/comp(Y,y,x,X2) by Th53;
hence thesis by Th53;
end;
::$N Associativity of Multiplication for Surreal Numbers, Conway Ch.1 Th.7(iii)
theorem
(x*y)*z == x*(y*z)
proof
defpred P[Ordinal] means
for x,y,z be Surreal st born x (+) born y (+) born z c= $1 holds
(x*y)*z == x*(y*z);
A1: for D be Ordinal st for C be Ordinal st C in D holds P[C] holds P[D]
proof
let D be Ordinal such that A2: for C be Ordinal st C in D holds P[C];
let x,y,z such that A3:born x (+) born y (+) born z c= D;
set xy=x*y, yz=y*z;
A4: xy = [comp(L_x,x,y,L_y) \/ comp(R_x,x,y,R_y),
comp(L_x,x,y,R_y) \/ comp(R_x,x,y,L_y)] by Th50;
A5: yz = [comp(L_y,y,z,L_z) \/ comp(R_y,y,z,R_z),
comp(L_y,y,z,R_z) \/ comp(R_y,y,z,L_z)] by Th50;
A6:x*yz = [comp(L_x,x,yz,L_yz) \/ comp(R_x,x,yz,R_yz),
comp(L_x,x,yz,R_yz) \/ comp(R_x,x,yz,L_yz)] by Th50;
A7:xy * z = [comp(L_xy,xy,z,L_z) \/ comp(R_xy,xy,z,R_z),
comp(L_xy,xy,z,R_z) \/ comp(R_xy,xy,z,L_z)] by Th50;
A8: for X,Y,Z be surreal-membered set st
(X=L_x or X=R_x) & (Y=L_y or Y=R_y) & (Z=L_z or Z=R_z)
holds comp(X,x,yz,comp(Y,y,z,Z)) <==> comp(comp(X,x,y,Y),xy,z,Z)
proof
let X,Y,Z be surreal-membered set such that
A9: (X=L_x or X=R_x) & (Y=L_y or Y=R_y) & (Z=L_z or Z=R_z);
for o be Surreal st o in comp(X,x,yz,comp(Y,y,z,Z))
ex b be Surreal st b in comp(comp(X,x,y,Y),xy,z,Z) & o==b
proof
let o be Surreal;
assume o in comp(X,x,yz,comp(Y,y,z,Z));
then consider x1,yz1 be Surreal such that
A10: o = x1*yz + x*yz1 - x1*yz1 & x1 in X & yz1 in comp(Y,y,z,Z)
by Def15;
consider y1,z1 be Surreal such that
A11: yz1 = y1*z + y*z1 - y1*z1 & y1 in Y & z1 in Z by A10,Def15;
A12:x*(y1*z + y*z1) == x*(y1*z) + x*(y*z1) by Th67;
A13:x*(y1*z + y*z1) + (x * (- y1*z1)) ==
(x*(y1*z) + x*(y*z1)) + (x * (- y1*z1)) by A12,Th43;
x*((y1*z + y*z1) + (- y1*z1)) == x*(y1*z + y*z1) + (x * (- y1*z1))
by Th67;
then A14:x*((y1*z + y*z1) + (- y1*z1)) ==
(x*(y1*z) + x*(y*z1)) + (x * (- y1*z1)) by A13,SURREALO:4;
A15:x1*(y1*z + y*z1) == x1*(y1*z) + x1*(y*z1) by Th67;
A16:x1*(y1*z + y*z1) + (x1 * (- y1*z1)) ==
(x1*(y1*z) + x1*(y*z1)) + (x1 * (- y1*z1)) by A15,Th43;
x1*((y1*z + y*z1) + (- y1*z1)) == x1*(y1*z + y*z1) + (x1 * (- y1*z1))
by Th67;
then x1*((y1*z + y*z1) + (- y1*z1)) ==
(x1*(y1*z) + x1*(y*z1)) + (x1 * (- y1*z1)) by A16,SURREALO:4;
then A17: -( x1*((y1*z + y*z1) + (- y1*z1))) ==
- ((x1*(y1*z) + x1*(y*z1)) + (x1 * (- y1*z1))) by Th10;
x1*yz + x*(y1*z + y*z1 + - y1*z1) ==
x1*yz +((x*(y1*z) + x*(y*z1)) + (x * (- y1*z1)))
by A14,Th43;
then A18: o == x1*yz +((x*(y1*z) + x*(y*z1)) + (x * (- y1*z1)))
+- ((x1*(y1*z) + x1*(y*z1)) + (x1 * (- y1*z1))) by A10,A11,A17,Th43;
A19: R_x<> {x} <> L_x & R_y<>{y} <> L_y & R_z<>{z} <> L_z
by SURREALO:2;
A20:x in {x} & y in {y} & z in {z} by TARSKI:def 1;
then born x1 (+) born y (+) born z in born x (+) born y (+) born z
by A10,A9,A19,Lm7;
then A21: x1*y*z == x1*(y*z) by A2,A3;
born x (+) born y1 (+) born z in born x (+) born y (+) born z
by A11,A9,A19,A20,Lm7;
then A22: x*y1*z == x*(y1*z) by A2,A3;
born x (+) born y (+) born z1 in born x (+) born y (+) born z
by A11,A9,A19,A20,Lm7;
then
A23: x*y*z1 == x*(y*z1) by A2,A3;
born x (+) born y1 (+) born z1 in born x (+) born y (+) born z
by A11,A9,A19,A20,Lm7;
then
A24: x*y1*z1 == x*(y1*z1) by A2,A3;
born x1 (+) born y (+) born z1 in born x (+) born y (+) born z
by A10,A11,A9,A19,A20,Lm7;
then
A25: x1*y*z1 == x1*(y*z1) by A2,A3;
born x1 (+) born y1 (+) born z in born x (+) born y (+) born z
by A10,A11,A9,A19,A20,Lm7;
then
A26: x1*y1*z == x1*(y1*z) by A2,A3;
born x1 (+) born y1 (+) born z1 in born x (+) born y (+) born z
by A10,A11,A9,A19,Lm7;
then
A27: x1*y1*z1 == x1*(y1*z1) by A2,A3;
A28: x1*yz +x*(y1*z) == x1*y*z + x*y1*z by A21,A22,Th43;
(x * (- y1*z1)) = - (x * (y1*z1)) by Th58;
then A29: (x * (- y1*z1)) == - (x * y1*z1) by A24,Th10;
(- x1*y1)*z = - ((x1*y1)*z) == - (x1*(y1*z)) by A26,Th58,Th10;
then (x1*yz +x*(y1*z)) + (- (x1*(y1*z))) ==
x1*y*z + x*y1*z + (- x1*y1)*z by A28,Th43;
then (x1*yz +x*(y1*z)) + (- (x1*(y1*z)))+ x*(y*z1) ==
x1*y*z + x*y1*z + (- x1*y1)*z + xy*z1 by Th43,A23;
then A30: (x1*yz +x*(y1*z)) + (-(x1*(y1*z)))+ x*(y*z1) + (x *(-y1*z1))
== x1*y*z + x*y1*z + (- x1*y1)*z + xy*z1 +-( x*y1*z1)
by Th43,A29;
x1 * (- y1*z1) = - (x1 * (y1*z1)) by Th58;
then x1 * (- y1*z1) == - (x1 * y1*z1) by Th10,A27;
then x1 * (- y1*z1) == (-x1 * y1)*z1 by Th58;
then (x1*(y*z1) + (x1 * (- y1*z1))) == (x1*y*z1 + (- x1*y1)*z1)
by Th43,A25;
then A31: -(x1*(y*z1) + (x1 * (- y1*z1))) == -(x1*y*z1+ (- x1*y1)*z1)
by Th10;
A32: (x1*yz +x*(y1*z)) + (- (x1*(y1*z))) + x*(y*z1) +
(x * (- y1*z1)) + - ((x1*(y*z1) + (x1 * (- y1*z1))))==
x1*y*z + x*y1*z + (- x1*y1)*z + xy*z1 +
-( x*y1*z1) + -(x1*y*z1+ (- x1*y1)*z1)
by A30,A31,Th43;
(x1*(y1*z) + x1*(y*z1)) + (x1 * (- y1*z1)) =
x1*(y1*z) + (x1*(y*z1) + (x1 * (- y1*z1))) by Th37;
then A33: - ((x1*(y1*z) + x1*(y*z1)) + (x1 * (- y1*z1))) =
- (x1*(y1*z)) + - ((x1*(y*z1) + (x1 * (- y1*z1))))
by Th40;
x1*y*z1 + x*y1*z1 +(- x1*y1)*z1 =
x*y1*z1 + (x1*y*z1+ (- x1*y1)*z1) by Th37;
then A34:-(x1*y*z1 + x*y1*z1 + (- x1*y1)*z1) =
-( x*y1*z1) + -(x1*y*z1+ (- x1*y1)*z1) by Th40;
A35: x1*yz +((x*(y1*z) + x*(y*z1)) + (x * (- y1*z1))) + (- (x1*(y1*z)))
= x1*yz +(x*(y1*z) + (x*(y*z1) + (x * (- y1*z1)))) + (- (x1*(y1*z)))
by Th37
.= (x1*yz +x*(y1*z)) + (x*(y*z1) + (x * (- y1*z1))) + (- (x1*(y1*z)))
by Th37
.= (x1*yz +x*(y1*z)) + ((x*(y*z1) + (x * (- y1*z1))) + (- (x1*(y1*z))))
by Th37
.= (x1*yz +x*(y1*z)) + ((- (x1*(y1*z))) + x*(y*z1) + (x * (- y1*z1)))
by Th37
.= (x1*yz +x*(y1*z)) + ((- (x1*(y1*z))) + x*(y*z1)) + (x * (- y1*z1))
by Th37
.= (x1*yz +x*(y1*z)) + (- (x1*(y1*z))) + x*(y*z1) + (x * (- y1*z1))
by Th37;
A36: x1*yz +((x*(y1*z) + x*(y*z1)) + (x * (- y1*z1))) +
(-(x1*(y1*z)) + - ((x1*(y*z1) + (x1 * (- y1*z1)))))
=
x1*yz +((x*(y1*z) + x*(y*z1)) + (x * (- y1*z1))) + -(x1*(y1*z))
+ - ((x1*(y*z1) + (x1 * (- y1*z1)))) by Th37;
x1*yz +((x*(y1*z) + x*(y*z1)) + (x * (- y1*z1)))
+- ((x1*(y1*z) + x1*(y*z1)) + (x1 * (- y1*z1)))==
x1*y*z + x*y1*z + (- x1*y1)*z + xy*z1 +
(-(x1*y*z1 + x*y1*z1 + (- x1*y1)*z1)) by A32,A35,A33,A34,A36,Th37;
then A37: o == x1*y*z + x*y1*z + (- x1*y1)*z + xy*z1 +
(-(x1*y*z1 + x*y1*z1 + (- x1*y1)*z1)) by A18,SURREALO:4;
(x1*y + x*y1)*z == x1*y*z + x*y1*z by Th67;
then A38:(x1*y + x*y1)*z + (- x1*y1)*z == x1*y*z + x*y1*z + (- x1*y1)*z
by Th43;
(x1*y + x*y1 + - x1*y1)*z == (x1*y + x*y1)*z + (- x1*y1)*z by Th67;
then A39: (x1*y + x*y1 + - x1*y1)*z == x1*y*z + x*y1*z + (- x1*y1)*z
by A38,SURREALO:4;
(x1*y + x*y1)*z1 == x1*y*z1 + x*y1*z1 by Th67;
then A40:(x1*y + x*y1)*z1 + (- x1*y1)*z1 ==
x1*y*z1 + x*y1*z1 + (- x1*y1)*z1 by Th43;
(x1*y + x*y1 + - x1*y1)*z1 == (x1*y + x*y1)*z1 + (- x1*y1)*z1 by Th67;
then (x1*y + x*y1 + - x1*y1)*z1 == x1*y*z1 + x*y1*z1 + (- x1*y1)*z1
by A40,SURREALO:4;
then A41: - ((x1*y + x*y1 + - x1*y1)*z1) ==
-(x1*y*z1 + x*y1*z1 + (- x1*y1)*z1) by Th10;
set xy1= x1*y + x*y1 - x1*y1;
xy1*z + xy*z1 == x1*y*z + x*y1*z + (- x1*y1)*z + xy*z1 by A39,Th43;
then xy1*z + xy*z1 +- xy1*z1 ==
x1*y*z + x*y1*z + (- x1*y1)*z + xy*z1 +
(-(x1*y*z1 + x*y1*z1 + (- x1*y1)*z1)) by Th43,A41;
then A42:o == xy1*z + xy*z1 - xy1*z1 by A37,SURREALO:4;
xy1 in comp(X,x,y,Y) by Def15,A10,A11;
then xy1*z + xy*z1 - xy1*z1 in comp(comp(X,x,y,Y),xy,z,Z)by Def15,A11;
hence thesis by A42;
end;
hence comp(X,x,yz,comp(Y,y,z,Z)) <=_ comp(comp(X,x,y,Y),xy,z,Z) by Th61;
for o be Surreal st o in comp(comp(X,x,y,Y),xy,z,Z)
ex b be Surreal st b in comp(X,x,yz,comp(Y,y,z,Z)) & o==b
proof
let o be Surreal;assume o in comp(comp(X,x,y,Y),xy,z,Z);
then consider xy1,z1 be Surreal such that
A43: o = xy1*z + xy*z1 - xy1*z1 & xy1 in comp(X,x,y,Y) & z1 in Z
by Def15;
consider x1,y1 be Surreal such that
A44: xy1 = x1*y + x*y1 - x1*y1 & x1 in X & y1 in Y by A43,Def15;
set yz1 = y1*z + y*z1 - y1*z1;
A45:x*(y1*z + y*z1) == x*(y1*z) + x*(y*z1) by Th67;
A46:x*(y1*z + y*z1) + (x * (- y1*z1)) ==
(x*(y1*z) + x*(y*z1)) + (x * (- y1*z1)) by A45,Th43;
x*((y1*z + y*z1) + (- y1*z1)) ==
x*(y1*z + y*z1) + (x * (- y1*z1)) by Th67;
then A47:x*((y1*z + y*z1) + (- y1*z1)) ==
(x*(y1*z) + x*(y*z1)) + (x * (- y1*z1)) by A46,SURREALO:4;
A48:x1*(y1*z + y*z1) == x1*(y1*z) + x1*(y*z1) by Th67;
A49:x1*(y1*z + y*z1) + (x1 * (- y1*z1)) ==
(x1*(y1*z) + x1*(y*z1)) + (x1 * (- y1*z1)) by A48,Th43;
x1*((y1*z + y*z1) + (- y1*z1)) == x1*(y1*z + y*z1) + (x1 * (- y1*z1))
by Th67;
then x1*((y1*z + y*z1) + (- y1*z1)) ==
(x1*(y1*z) + x1*(y*z1)) + (x1 * (- y1*z1)) by A49,SURREALO:4;
then A50: -( x1*((y1*z + y*z1) + (- y1*z1))) ==
- ((x1*(y1*z) + x1*(y*z1)) + (x1 * (- y1*z1))) by Th10;
x1*yz + x*(y1*z + y*z1 + - y1*z1) ==
x1*yz +((x*(y1*z) + x*(y*z1)) + (x * (- y1*z1))) by A47,Th43;
then A51: x1*yz + x*yz1 + - x1*yz1 ==
x1*yz +((x*(y1*z) + x*(y*z1)) + (x * (- y1*z1)))
+- ((x1*(y1*z) + x1*(y*z1)) + (x1 * (- y1*z1))) by A50,Th43;
A52: R_x<> {x} <> L_x & R_y<>{y} <> L_y & R_z<>{z} <> L_z
by SURREALO:2;
A53:x in {x} & y in {y} & z in {z} by TARSKI:def 1;
born x1 (+) born y (+) born z in born x (+) born y (+) born z
by A44,A9,A52,A53,Lm7;
then A54: x1*y*z == x1*(y*z) by A2,A3;
born x (+) born y1 (+) born z in born x (+) born y (+) born z
by A44,A9,A52,A53,Lm7;
then A55: x*y1*z == x*(y1*z) by A2,A3;
born x (+) born y (+) born z1 in born x (+) born y (+) born z
by A43,A9,A52,A53,Lm7;
then A56: x*y*z1 == x*(y*z1) by A2,A3;
born x (+) born y1 (+) born z1 in born x (+) born y (+) born z
by A43,A44,A9,A52,A53,Lm7;
then
A57: x*y1*z1 == x*(y1*z1) by A2,A3;
born x1 (+) born y (+) born z1 in born x (+) born y (+) born z
by A43,A44,A9,A52,A53,Lm7;
then
A58: x1*y*z1 == x1*(y*z1) by A2,A3;
born x1 (+) born y1 (+) born z in born x (+) born y (+) born z
by A44,A9,A52,A53,Lm7;
then
A59: x1*y1*z == x1*(y1*z) by A2,A3;
born x1 (+) born y1 (+) born z1 in born x (+) born y (+) born z
by A43,A44,A9,A52,Lm7;
then
A60: x1*y1*z1 == x1*(y1*z1) by A2,A3;
A61: x1*yz +x*(y1*z) == x1*y*z + x*y1*z by A54,A55,Th43;
(x * (- y1*z1)) = - (x * (y1*z1)) by Th58;
then A62: (x * (- y1*z1)) == - (x * y1*z1) by A57,Th10;
(- x1*y1)*z = - ((x1*y1)*z) == - (x1*(y1*z)) by A59,Th58,Th10;
then (x1*yz +x*(y1*z)) + (- (x1*(y1*z))) ==
x1*y*z + x*y1*z + (- x1*y1)*z by A61,Th43;
then (x1*yz +x*(y1*z)) + (- (x1*(y1*z)))+ x*(y*z1) ==
x1*y*z + x*y1*z + (- x1*y1)*z + xy*z1 by Th43,A56;
then A63: (x1*yz +x*(y1*z)) + (- (x1*(y1*z)))+ x*(y*z1) +
(x * (- y1*z1)) == x1*y*z + x*y1*z + (- x1*y1)*z + xy*z1 +-( x*y1*z1)
by Th43,A62;
x1 * (- y1*z1) = - (x1 * (y1*z1)) by Th58;
then x1 * (- y1*z1) == - (x1 * y1*z1) by Th10,A60;
then x1 * (- y1*z1) == (-x1 * y1)*z1 by Th58;
then (x1*(y*z1) + (x1 * (- y1*z1))) == (x1*y*z1 + (- x1*y1)*z1)
by Th43,A58;
then A64: -(x1*(y*z1) + (x1 * (- y1*z1))) == -(x1*y*z1+ (- x1*y1)*z1)
by Th10;
(x1*(y1*z) + x1*(y*z1)) + (x1 * (- y1*z1)) =
x1*(y1*z) + (x1*(y*z1) + (x1 * (- y1*z1))) by Th37;
then A65: - ((x1*(y1*z) + x1*(y*z1)) + (x1 * (- y1*z1))) =
- (x1*(y1*z)) + - ((x1*(y*z1) + (x1 * (- y1*z1)))) by Th40;
x1*y*z1 + x*y1*z1 + (- x1*y1)*z1 =
x*y1*z1 + (x1*y*z1+ (- x1*y1)*z1) by Th37;
then A66:-(x1*y*z1 + x*y1*z1 + (- x1*y1)*z1) =
-( x*y1*z1) + -(x1*y*z1+ (- x1*y1)*z1) by Th40;
A67: x1*yz +((x*(y1*z) + x*(y*z1)) + (x * (- y1*z1))) + (- (x1*(y1*z)))
= x1*yz +(x*(y1*z) + (x*(y*z1) + (x * (- y1*z1)))) + (- (x1*(y1*z)))
by Th37
.= (x1*yz +x*(y1*z)) + (x*(y*z1) + (x * (- y1*z1))) + (- (x1*(y1*z)))
by Th37
.= (x1*yz +x*(y1*z)) + ((x*(y*z1) + (x * (- y1*z1))) + (- (x1*(y1*z))))
by Th37
.= (x1*yz +x*(y1*z)) + ((- (x1*(y1*z))) + x*(y*z1) + (x * (- y1*z1)))
by Th37
.= (x1*yz +x*(y1*z)) + ((- (x1*(y1*z))) + x*(y*z1)) + (x * (- y1*z1))
by Th37
.= (x1*yz +x*(y1*z)) + (- (x1*(y1*z))) + x*(y*z1) + (x * (- y1*z1))
by Th37;
A68: x1*yz +((x*(y1*z) + x*(y*z1)) + (x * (- y1*z1))) + (-(x1*(y1*z)) +
- ((x1*(y*z1) + (x1 * (- y1*z1))))) =
x1*yz +((x*(y1*z) + x*(y*z1)) + (x * (- y1*z1))) + -(x1*(y1*z))+
- ((x1*(y*z1) + (x1 * (- y1*z1)))) by Th37;
x1*yz +((x*(y1*z) + x*(y*z1)) + (x * (- y1*z1))) + -(x1*(y1*z))+
- ((x1*(y*z1) + (x1 * (- y1*z1))))==
x1*y*z + x*y1*z + (- x1*y1)*z + xy*z1 +-( x*y1*z1)+
-(x1*y*z1+ (- x1*y1)*z1) by A63,A64,Th43,A67;
then x1*yz +((x*(y1*z) + x*(y*z1)) + (x * (- y1*z1))) +-
((x1*(y1*z) + x1*(y*z1)) + (x1 * (- y1*z1)))==
x1*y*z + x*y1*z + (- x1*y1)*z + xy*z1 + (-(x1*y*z1 + x*y1*z1
+ (- x1*y1)*z1)) by A65,A66,A68,Th37;
then A69: x1*yz + x*yz1 + - x1*yz1 ==
x1*y*z + x*y1*z + (- x1*y1)*z + xy*z1
+ (-(x1*y*z1 + x*y1*z1 + (- x1*y1)*z1)) by A51,SURREALO:4;
(x1*y + x*y1)*z == x1*y*z + x*y1*z by Th67;
then A70:(x1*y + x*y1)*z + (- x1*y1)*z == x1*y*z + x*y1*z + (- x1*y1)*z
by Th43;
(x1*y + x*y1 + - x1*y1)*z == (x1*y + x*y1)*z + (- x1*y1)*z by Th67;
then A71: (x1*y + x*y1 + - x1*y1)*z == x1*y*z + x*y1*z + (- x1*y1)*z
by A70,SURREALO:4;
(x1*y + x*y1)*z1 == x1*y*z1 + x*y1*z1 by Th67;
then A72:(x1*y + x*y1)*z1 + (- x1*y1)*z1 ==
x1*y*z1 + x*y1*z1 + (- x1*y1)*z1 by Th43;
(x1*y + x*y1 + - x1*y1)*z1 == (x1*y + x*y1)*z1 + (- x1*y1)*z1 by Th67;
then (x1*y + x*y1 + - x1*y1)*z1 == x1*y*z1 + x*y1*z1 + (- x1*y1)*z1
by A72,SURREALO:4;
then A73: - ((x1*y + x*y1 + - x1*y1)*z1) ==
-(x1*y*z1 + x*y1*z1 + (- x1*y1)*z1) by Th10;
xy1*z + xy*z1 == x1*y*z + x*y1*z + (-x1*y1)*z + xy*z1 by A44,A71,Th43;
then xy1*z + xy*z1 +- xy1*z1 == x1*y*z + x*y1*z + (- x1*y1)*z + xy*z1 +
(-(x1*y*z1 + x*y1*z1 + (- x1*y1)*z1)) by A44,Th43,A73;
then A74:o == x1*yz + x*yz1 - x1*yz1 by A43,A69,SURREALO:4;
yz1 in comp(Y,y,z,Z) by Def15,A43,A44;
then x1*yz + x*yz1 - x1*yz1 in comp(X,x,yz,comp(Y,y,z,Z))
by Def15,A44;
hence thesis by A74;
end;
hence thesis by Th61;
end;
A75:comp(L_xy,xy,z,L_z)
= comp(comp(L_x,x,y,L_y),xy,z,L_z)\/comp(comp(R_x,x,y,R_y),xy,z,L_z)
by A4,Th68;
A76: comp(R_xy,xy,z,R_z) =
comp(comp(L_x,x,y,R_y),xy,z,R_z) \/ comp(comp(R_x,x,y,L_y),xy,z,R_z)
by A4,Th68;
A77: comp(L_x,x,yz,L_yz)
= comp(L_x,x,yz,comp(L_y,y,z,L_z)) \/ comp(L_x,x,yz,comp(R_y,y,z,R_z))
by A5,Th60;
A78: comp(R_x,x,yz,R_yz)
= comp(R_x,x,yz,comp(L_y,y,z,R_z)) \/ comp(R_x,x,yz,comp(R_y,y,z,L_z))
by A5,Th60;
A79:comp(comp(L_x,x,y,L_y),xy,z,L_z) <==> comp(L_x,x,yz,comp(L_y,y,z,L_z))
by A8;
A80:comp(comp(R_x,x,y,R_y),xy,z,L_z) <==> comp(R_x,x,yz,comp(R_y,y,z,L_z))
by A8;
A81:comp(comp(L_x,x,y,R_y),xy,z,R_z) <==> comp(L_x,x,yz,comp(R_y,y,z,R_z))
by A8;
A82:comp(comp(R_x,x,y,L_y),xy,z,R_z) <==> comp(R_x,x,yz,comp(L_y,y,z,R_z))
by A8;
A83:comp(L_xy,xy,z,L_z) <==> comp(L_x,x,yz,comp(L_y,y,z,L_z))
\/ comp(R_x,x,yz,comp(R_y,y,z,L_z))
by A75,A79,A80,SURREALO:31;
A84: comp(R_xy,xy,z,R_z) <==> comp(L_x,x,yz,comp(R_y,y,z,R_z))
\/ comp(R_x,x,yz,comp(L_y,y,z,R_z))
by A76,A81,A82,SURREALO:31;
A85: (comp(L_x,x,yz,comp(L_y,y,z,L_z))\/ comp(R_x,x,yz,comp(R_y,y,z,L_z)))
\/ (comp(L_x,x,yz,comp(R_y,y,z,R_z))\/ comp(R_x,x,yz,comp(L_y,y,z,R_z)))
= ((comp(L_x,x,yz,comp(L_y,y,z,L_z))\/ comp(R_x,x,yz,comp(R_y,y,z,L_z))) \/
comp(L_x,x,yz,comp(R_y,y,z,R_z)) )\/ comp(R_x,x,yz,comp(L_y,y,z,R_z))
by XBOOLE_1:4
.= ((comp(L_x,x,yz,comp(L_y,y,z,L_z))\/ comp(L_x,x,yz,comp(R_y,y,z,R_z)))\/
comp(R_x,x,yz,comp(R_y,y,z,L_z))) \/ comp(R_x,x,yz,comp(L_y,y,z,R_z))
by XBOOLE_1:4
.= (comp(L_x,x,yz,comp(L_y,y,z,L_z)) \/ comp(L_x,x,yz,comp(R_y,y,z,R_z)))\/
(comp(R_x,x,yz,comp(L_y,y,z,R_z)) \/ comp(R_x,x,yz,comp(R_y,y,z,L_z)))
by XBOOLE_1:4;
A86: L_(xy * z) <==> L_(x*yz) by A6,A85,A77,A78,A84,A7,A83,
SURREALO:31;
A87: comp(L_x,x,yz,R_yz)=
comp(L_x,x,yz,comp(L_y,y,z,R_z))\/comp(L_x,x,yz,comp(R_y,y,z,L_z))
by A5,Th60;
A88:comp(R_x,x,yz,L_yz) =
comp(R_x,x,yz,comp(L_y,y,z,L_z)) \/comp(R_x,x,yz,comp(R_y,y,z,R_z))
by A5,Th60;
A89: comp(L_x,x,yz,comp(L_y,y,z,R_z)) <==>
comp(comp(L_x,x,y,L_y),xy,z,R_z) by A8;
A90: comp(L_x,x,yz,comp(R_y,y,z,L_z)) <==>
comp(comp(L_x,x,y,R_y),xy,z,L_z) by A8;
A91:comp(R_x,x,yz,comp(L_y,y,z,L_z)) <==>
comp(comp(R_x,x,y,L_y),xy,z,L_z) by A8;
A92:comp(R_x,x,yz,comp(R_y,y,z,R_z)) <==>
comp(comp(R_x,x,y,R_y),xy,z,R_z) by A8;
A93: comp(L_x,x,yz,R_yz) <==>
comp(comp(L_x,x,y,L_y),xy,z,R_z)\/comp(comp(L_x,x,y,R_y),xy,z,L_z)
by A87,A89,A90,SURREALO:31;
comp(R_x,x,yz,L_yz) <==>
comp(comp(R_x,x,y,L_y),xy,z,L_z)\/ comp(comp(R_x,x,y,R_y),xy,z,R_z)
by A88,A91,A92,SURREALO:31;
then
A94: R_(x*yz) <==>
(comp(comp(L_x,x,y,L_y),xy,z,R_z)\/comp(comp(L_x,x,y,R_y),xy,z,L_z))\/
(comp(comp(R_x,x,y,L_y),xy,z,L_z)\/ comp(comp(R_x,x,y,R_y),xy,z,R_z))
by A6,A93,SURREALO:31;
A95: (comp(comp(L_x,x,y,L_y),xy,z,R_z)\/comp(comp(L_x,x,y,R_y),xy,z,L_z))\/
(comp(comp(R_x,x,y,L_y),xy,z,L_z)\/ comp(comp(R_x,x,y,R_y),xy,z,R_z))
=
((comp(comp(L_x,x,y,L_y),xy,z,R_z)\/comp(comp(L_x,x,y,R_y),xy,z,L_z))\/
comp(comp(R_x,x,y,L_y),xy,z,L_z))\/ comp(comp(R_x,x,y,R_y),xy,z,R_z)
by XBOOLE_1:4
.=( comp(comp(L_x,x,y,L_y),xy,z,R_z)\/
(comp(comp(L_x,x,y,R_y),xy,z,L_z)\/
comp(comp(R_x,x,y,L_y),xy,z,L_z)))\/ comp(comp(R_x,x,y,R_y),xy,z,R_z)
by XBOOLE_1:4
.= (comp(comp(L_x,x,y,L_y),xy,z,R_z)\/comp(comp(R_x,x,y,R_y),xy,z,R_z))\/
(comp(comp(L_x,x,y,R_y),xy,z,L_z)\/comp(comp(R_x,x,y,L_y),xy,z,L_z))
by XBOOLE_1:4;
A96: comp(L_xy,xy,z,R_z) =
comp(comp(L_x,x,y,L_y),xy,z,R_z)\/comp(comp(R_x,x,y,R_y),xy,z,R_z)
by A4,Th68;
comp(R_xy,xy,z,L_z) =
comp(comp(L_x,x,y,R_y),xy,z,L_z)\/comp(comp(R_x,x,y,L_y),xy,z,L_z)
by A4,Th68;
hence thesis by A6,A7,A86,SURREALO:29,A94,A95,A96;
end;
for D be Ordinal holds P[D] from ORDINAL1:sch 2(A1);
then P[born x (+) born y (+) born z];
hence thesis;
end;
theorem Th70:
0_No < x & y < z implies y*x < z*x
proof
assume 0_No < x & y < z;
then 0_No*z+x*y < 0_No*y+x*z by Th51;
hence thesis;
end;
theorem
x < 0_No & y < z implies z*x < y*x
proof
assume x <0_No & y < z;
then x*z+0_No*y < x*y+0_No*z by Th51;
hence thesis;
end;
Lm9:
0_No < x & 0_No < y implies 0_No < x*y
proof
assume 0_No < x & 0_No < y;
then 0_No*y + x *0_No < 0_No*0_No+x*y by Th51;
hence thesis;
end;
::$N Conway Ch.1 Th.9
theorem Th72:
0_No < x*y iff (x < 0_No & y < 0_No) or (0_No < x & 0_No < y)
proof
thus 0_No < x*y implies (x < 0_No & y < 0_No) or (0_No < x & 0_No < y)
proof
assume that A1:0_No < x*y and
A2:not (x < 0_No & y < 0_No) & not (0_No < x & 0_No < y);
A3:0_No <= x*y by A1;
A4:0_No < y or y < 0_No
proof
assume not (0_No < y or y < 0_No);
then 0_No==y;
then 0_No * x == x*y by Th51;
hence thesis by A1;
end;
per cases;
suppose x==0_No;
then 0_No * y == x*y by Th51;
hence thesis by A1;
end;
suppose A5: 0_No < x;
then -0_No < -y by A2,A4,Th10;
then -0_No = 0_No < x*(-y)= -(x*y) by A5,Lm9,Th58;
hence thesis by A3,Th10;
end;
suppose A6:x < 0_No;
then 0_No < -x by Th10,Th23;
then -0_No= 0_No < (-x)*y = -(x*y) by A6,A2,A4,Lm9,Th58;
hence thesis by Th10,A3;
end;
end;
assume (x < 0_No & y < 0_No) or (0_No < x & 0_No < y);
then per cases;
suppose x < 0_No & y < 0_No;
then -0_No < -x & -0_No < -y & -0_No = 0_No by Th10;
then 0_No < (-x)*(-y) = - (-x)*y by Lm9,Th58;
hence thesis by Th58;
end;
suppose 0_No < x & 0_No < y;
hence thesis by Lm9;
end;
end;
theorem
0_No < z & x*z < y*z implies x < y
proof
assume that A1: 0_No < z and A2:x*z < y*z;
assume y <=x;
then per cases;
suppose y==x;
then y*z==x*z by Th51;
hence thesis by A2;
end;
suppose y < x;
then y*z <= x*z by A1,Th70;
hence thesis by A2;
end;
end;
theorem Th74:
x*y < 0_No iff (x < 0_No & 0_No < y ) or (0_No < x & y < 0_No)
proof
set z=-y;
thus x*y < 0_No implies (x < 0_No & 0_No < y ) or (0_No < x & y < 0_No)
proof
assume x*y < 0_No;
then 0_No < -(x*y)=x*z by Th23,Th10,Th58;
then (x < 0_No & z < 0_No) or (0_No < x & 0_No < z) by Th72;
hence thesis by Th23,Th10;
end;
assume (x < 0_No & 0_No < y ) or (0_No < x & y < 0_No);
then (x < 0_No & z < 0_No) or (0_No < x & 0_No < z) by Th23,Th10;
then 0_No < x*z = - (x*y) by Th72,Th58;
hence thesis by Th23,Th10;
end;
theorem
0_No <= x & y <= z implies y*x <= z*x
proof
assume A1: 0_No <= x & y <= z;
per cases;
suppose x <= 0_No;
then x == 0_No by A1;
then y*x == y * 0_No = 0_No = z * 0_No == z*x by Th51;
hence thesis by SURREALO:4;
end;
suppose A2: 0_No < x;
per cases;
suppose z <= y;
then z==y by A1;
then y*x == z*x by Th51;
hence thesis;
end;
suppose y < z;
hence thesis by A2,Th70;
end;
end;
end;
theorem
(x+y)*(x+y) == x*x + y*y +(x*y + y*x)
proof
x*(x+y) == x*x + x*y & y*(x+y) == y*x + y*y by Th67;
then A1: (x+y)*(x+y) == x*(x+y) + y*(x+y) == (x*x+x*y) + (y*x + y*y)
by Th67,Th43;
(x*x+x*y) + (y*x + y*y) = x*x+(x*y + (y*x + y*y)) by Th37
.= x*x+(x*y + y*x + y*y) by Th37
.= x*x+y*y+(x*y + y*x) by Th37;
hence thesis by A1,SURREALO:4;
end;
theorem
x*y ==0_No iff x == 0_No or y == 0_No by Th72,Th74;