euclid_8.miz



    begin

    reserve r,r1,r2,x,y,z,x1,x2,x3,y1,y2,y3 for Real;

    reserve R,R1,R2,R3 for Element of (3 -tuples_on REAL );

    reserve p,q,p1,p2,p3,q1,q2 for Element of ( REAL 3);

    reserve f1,f2,f3,g1,g2,g3,h1,h2,h3 for PartFunc of REAL , REAL ;

    reserve t,t0,t1,t2 for Real;

    definition

      let x,y,z be Real;

      :: original: |[

      redefine

      func |[x,y,z]| -> Element of ( REAL 3) ;

      coherence

      proof

        reconsider x, y, z as Element of REAL by XREAL_0:def 1;

         <*x, y, z*> is Element of ( REAL 3) by FINSEQ_2: 104;

        hence thesis;

      end;

    end

    theorem :: EUCLID_8:1

    

     Th1: p = |[(p . 1), (p . 2), (p . 3)]|

    proof

      consider x,y,z be Element of REAL such that

       A1: p = <*x, y, z*> by FINSEQ_2: 103;

      

       A2: (p . 1) = x by A1, FINSEQ_1: 45;

      (p . 2) = y by A1, FINSEQ_1: 45;

      hence thesis by A1, A2, FINSEQ_1: 45;

    end;

    

     Lm1: for r be Real holds (r * p) = |[(r * (p . 1)), (r * (p . 2)), (r * (p . 3))]|

    proof

      let r be Real;

      

       A1: ((r * p) . 1) = (r * (p . 1)) by RVSUM_1: 44;

      

       A2: ((r * p) . 2) = (r * (p . 2)) by RVSUM_1: 44;

      ((r * p) . 3) = (r * (p . 3)) by RVSUM_1: 44;

      hence thesis by A1, A2, Th1;

    end;

    

     Lm2: (p1 + p2) = |[((p1 . 1) + (p2 . 1)), ((p1 . 2) + (p2 . 2)), ((p1 . 3) + (p2 . 3))]|

    proof

      

       A1: ((p1 + p2) . 1) = ((p1 . 1) + (p2 . 1)) by RVSUM_1: 11;

      

       A2: ((p1 + p2) . 2) = ((p1 . 2) + (p2 . 2)) by RVSUM_1: 11;

      ((p1 + p2) . 3) = ((p1 . 3) + (p2 . 3)) by RVSUM_1: 11;

      hence thesis by A1, A2, Th1;

    end;

    

     Lm3: (( - p) . 1) = ( - (p . 1)) & (( - p) . 2) = ( - (p . 2)) & (( - p) . 3) = ( - (p . 3))

    proof

      ( - p) = |[(( - 1) * (p . 1)), (( - 1) * (p . 2)), (( - 1) * (p . 3))]| by Lm1

      .= |[( - (p . 1)), ( - (p . 2)), ( - (p . 3))]|;

      hence thesis by FINSEQ_1: 45;

    end;

    theorem :: EUCLID_8:2

    for f be FinSequence of REAL st ( len f) = 3 holds f is Element of ( REAL 3)

    proof

      let f be FinSequence of REAL ;

      assume

       A1: ( len f) = 3;

      reconsider x1 = (f . 1), x2 = (f . 2), x3 = (f . 3) as Element of REAL by XREAL_0:def 1;

       <*x1, x2, x3*> is Element of (3 -tuples_on REAL ) by FINSEQ_2: 104;

      hence thesis by A1, FINSEQ_1: 45;

    end;

    

     Lm4: (p1 - p2) = |[((p1 . 1) - (p2 . 1)), ((p1 . 2) - (p2 . 2)), ((p1 . 3) - (p2 . 3))]|

    proof

      

       A1: (( - p2) . 1) = ( - (p2 . 1)) by Lm3;

      

       A2: (( - p2) . 2) = ( - (p2 . 2)) by Lm3;

      (( - p2) . 3) = ( - (p2 . 3)) by Lm3;

      then (p1 + ( - p2)) = |[((p1 . 1) + ( - (p2 . 1))), ((p1 . 2) + ( - (p2 . 2))), ((p1 . 3) + ( - (p2 . 3)))]| by A1, A2, Lm2;

      hence thesis;

    end;

    

     Lm5: |(p1, p2)| = ((((p1 . 1) * (p2 . 1)) + ((p1 . 2) * (p2 . 2))) + ((p1 . 3) * (p2 . 3)))

    proof

      reconsider f1 = p1, f2 = p2 as FinSequence of REAL ;

      

       A1: ( len f1) = ( len <*(p1 . 1), (p1 . 2), (p1 . 3)*>) by Th1

      .= 3 by FINSEQ_1: 45;

      ( len f2) = ( len <*(p2 . 1), (p2 . 2), (p2 . 3)*>) by Th1

      .= 3 by FINSEQ_1: 45;

      

      then |(p1, p2)| = ( Sum <*((f1 . 1) * (f2 . 1)), ((f1 . 2) * (f2 . 2)), ((f1 . 3) * (f2 . 3))*>) by A1, EUCLID_5: 28

      .= ((((p1 . 1) * (p2 . 1)) + ((p1 . 2) * (p2 . 2))) + ((p1 . 3) * (f2 . 3))) by RVSUM_1: 78;

      hence thesis;

    end;

    

     Lm6: (r * |[x, y, z]|) = |[(r * x), (r * y), (r * z)]|

    proof

      set p = |[x, y, z]|;

      (r * p) = |[(r * (p . 1)), (r * (p . 2)), (r * (p . 3))]| by Lm1

      .= |[(r * x), (r * (p . 2)), (r * (p . 3))]| by FINSEQ_1: 45

      .= |[(r * x), (r * y), (r * (p . 3))]| by FINSEQ_1: 45

      .= |[(r * x), (r * y), (r * z)]| by FINSEQ_1: 45;

      hence thesis;

    end;

    

     Lm7: (p1 + ( - p2)) = |[((p1 . 1) - (p2 . 1)), ((p1 . 2) - (p2 . 2)), ((p1 . 3) - (p2 . 3))]|

    proof

      

       A1: ( - p2) = |[(( - 1) * (p2 . 1)), (( - 1) * (p2 . 2)), (( - 1) * (p2 . 3))]| by Lm1

      .= |[( - (p2 . 1)), ( - (p2 . 2)), ( - (p2 . 3))]|;

      (p1 + ( - p2)) = |[((p1 . 1) + (( - p2) . 1)), ((p1 . 2) + (( - p2) . 2)), ((p1 . 3) + (( - p2) . 3))]| by Lm2

      .= |[((p1 . 1) + ( - (p2 . 1))), ((p1 . 2) + (( - p2) . 2)), ((p1 . 3) + (( - p2) . 3))]| by A1, FINSEQ_1: 45

      .= |[((p1 . 1) + ( - (p2 . 1))), ((p1 . 2) + ( - (p2 . 2))), ((p1 . 3) + (( - p2) . 3))]| by A1, FINSEQ_1: 45

      .= |[((p1 . 1) + ( - (p2 . 1))), ((p1 . 2) + ( - (p2 . 2))), ((p1 . 3) + ( - (p2 . 3)))]| by A1, FINSEQ_1: 45;

      hence thesis;

    end;

    

     Lm8: ( |[x1, x2, x3]| + |[y1, y2, y3]|) = |[(x1 + y1), (x2 + y2), (x3 + y3)]|

    proof

      

       A1: ( |[y1, y2, y3]| . 1) = y1 by FINSEQ_1: 45;

      

       A2: ( |[y1, y2, y3]| . 2) = y2 by FINSEQ_1: 45;

      

       A3: ( |[y1, y2, y3]| . 3) = y3 by FINSEQ_1: 45;

      

       A4: (( |[x1, x2, x3]| . 1) + ( |[y1, y2, y3]| . 1)) = (x1 + y1) by A1, FINSEQ_1: 45;

      

       A5: (( |[x1, x2, x3]| . 2) + ( |[y1, y2, y3]| . 2)) = (x2 + y2) by A2, FINSEQ_1: 45;

      (( |[x1, x2, x3]| . 3) + ( |[y1, y2, y3]| . 3)) = (x3 + y3) by A3, FINSEQ_1: 45;

      hence thesis by A4, A5, Lm2;

    end;

    

     Lm9: ((p1 + p2) + (q1 + q2)) = ((p1 + q1) + (p2 + q2))

    proof

      

      thus ((p1 + p2) + (q1 + q2)) = (((p1 + p2) + q1) + q2) by RVSUM_1: 15

      .= (((p1 + q1) + p2) + q2) by RVSUM_1: 15

      .= ((p1 + q1) + (p2 + q2)) by RVSUM_1: 15;

    end;

    

     Lm10: ((p1 + p2) - (q1 + q2)) = ((p1 - q1) + (p2 - q2))

    proof

      

      thus ((p1 + p2) - (q1 + q2)) = (((p1 + p2) - q1) - q2) by RVSUM_1: 39

      .= ((p1 + p2) + (( - q1) + ( - q2))) by RVSUM_1: 15

      .= ((p1 - q1) + (p2 - q2)) by Lm9;

    end;

    

     Lm11: ( |[x1, x2, x3]| - |[y1, y2, y3]|) = |[(x1 - y1), (x2 - y2), (x3 - y3)]|

    proof

      

       A1: ( |[y1, y2, y3]| . 1) = y1 by FINSEQ_1: 45;

      

       A2: ( |[y1, y2, y3]| . 2) = y2 by FINSEQ_1: 45;

      

       A3: ( |[y1, y2, y3]| . 3) = y3 by FINSEQ_1: 45;

      

       A4: (( |[x1, x2, x3]| . 1) - ( |[y1, y2, y3]| . 1)) = (x1 - y1) by A1, FINSEQ_1: 45;

      

       A5: (( |[x1, x2, x3]| . 2) - ( |[y1, y2, y3]| . 2)) = (x2 - y2) by A2, FINSEQ_1: 45;

      (( |[x1, x2, x3]| . 3) - ( |[y1, y2, y3]| . 3)) = (x3 - y3) by A3, FINSEQ_1: 45;

      hence thesis by A4, A5, Lm7;

    end;

    definition

      :: EUCLID_8:def1

      func <e1> -> Element of ( REAL 3) equals |[1, 0 , 0 ]|;

      coherence ;

      :: EUCLID_8:def2

      func <e2> -> Element of ( REAL 3) equals |[ 0 , 1, 0 ]|;

      coherence ;

      :: EUCLID_8:def3

      func <e3> -> Element of ( REAL 3) equals |[ 0 , 0 , 1]|;

      coherence ;

    end

    

     Lm12: p = |[x, y, z]| implies |(p, p)| = (((x ^2 ) + (y ^2 )) + (z ^2 ))

    proof

      assume p = |[x, y, z]|;

      then (p . 1) = x & (p . 2) = y & (p . 3) = z by FINSEQ_1: 45;

      hence thesis by Lm5;

    end;

    definition

      let p1, p2;

      :: EUCLID_8:def4

      func p1 <X> p2 -> Element of ( REAL 3) equals |[(((p1 . 2) * (p2 . 3)) - ((p1 . 3) * (p2 . 2))), (((p1 . 3) * (p2 . 1)) - ((p1 . 1) * (p2 . 3))), (((p1 . 1) * (p2 . 2)) - ((p1 . 2) * (p2 . 1)))]|;

      correctness ;

    end

    

     Lm13: ( |[x1, x2, x3]| <X> |[y1, y2, y3]|) = |[((x2 * y3) - (x3 * y2)), ((x3 * y1) - (x1 * y3)), ((x1 * y2) - (x2 * y1))]|

    proof

      set p1 = |[x1, x2, x3]|;

      

       A1: (p1 . 1) = x1 & (p1 . 2) = x2 & (p1 . 3) = x3 by FINSEQ_1: 45;

      set p2 = |[y1, y2, y3]|;

      (p2 . 1) = y1 & (p2 . 2) = y2 & (p2 . 3) = y3 by FINSEQ_1: 45;

      hence thesis by A1;

    end;

    

     Lm14: ((r * p1) <X> p2) = (r * (p1 <X> p2)) & ((r * p1) <X> p2) = (p1 <X> (r * p2))

    proof

      

       A1: ((p1 <X> p2) . 1) = (((p1 . 2) * (p2 . 3)) - ((p1 . 3) * (p2 . 2))) & ((p1 <X> p2) . 2) = (((p1 . 3) * (p2 . 1)) - ((p1 . 1) * (p2 . 3))) & ((p1 <X> p2) . 3) = (((p1 . 1) * (p2 . 2)) - ((p1 . 2) * (p2 . 1))) by FINSEQ_1: 45;

      

       A2: ((r * p1) <X> p2) = ( |[(r * (p1 . 1)), (r * (p1 . 2)), (r * (p1 . 3))]| <X> p2) by Lm1

      .= ( |[(r * (p1 . 1)), (r * (p1 . 2)), (r * (p1 . 3))]| <X> |[(p2 . 1), (p2 . 2), (p2 . 3)]|) by Th1

      .= |[(((r * (p1 . 2)) * (p2 . 3)) - ((r * (p1 . 3)) * (p2 . 2))), (((r * (p1 . 3)) * (p2 . 1)) - ((r * (p1 . 1)) * (p2 . 3))), (((r * (p1 . 1)) * (p2 . 2)) - ((r * (p1 . 2)) * (p2 . 1)))]| by Lm13;

      

      then

       A3: ((r * p1) <X> p2) = |[(r * ((p1 <X> p2) . 1)), (r * ((p1 <X> p2) . 2)), (r * ((p1 <X> p2) . 3))]| by A1

      .= (r * (p1 <X> p2)) by Lm1;

      ((r * p1) <X> p2) = |[(((p1 . 2) * (r * (p2 . 3))) - ((p1 . 3) * (r * (p2 . 2)))), (((p1 . 3) * (r * (p2 . 1))) - ((p1 . 1) * (r * (p2 . 3)))), (((p1 . 1) * (r * (p2 . 2))) - ((p1 . 2) * (r * (p2 . 1))))]| by A2

      .= ( |[(p1 . 1), (p1 . 2), (p1 . 3)]| <X> |[(r * (p2 . 1)), (r * (p2 . 2)), (r * (p2 . 3))]|) by Lm13

      .= (p1 <X> |[(r * (p2 . 1)), (r * (p2 . 2)), (r * (p2 . 3))]|) by Th1

      .= (p1 <X> (r * p2)) by Lm1;

      hence thesis by A3;

    end;

    theorem :: EUCLID_8:3

    (p1,p2) are_ldependent2 implies (p1 <X> p2) = ( 0.REAL 3)

    proof

      assume (p1,p2) are_ldependent2 ;

      then

       A1: ex a1,a2 be Real st ((a1 * p1) + (a2 * p2)) = ( 0.REAL 3) & (a1 <> 0 or a2 <> 0 ) by EUCLIDLP:def 2;

      now

        per cases by A1;

          case ex a1,a2 be Real st ((a1 * p1) + (a2 * p2)) = ( 0.REAL 3) & a1 <> 0 ;

          then

          consider a1,a2 be Real such that

           A2: a1 <> 0 & ((a1 * p1) + (a2 * p2)) = ( 0.REAL 3);

          

           A3: ((1 / a1) * (a1 * p1)) = ((1 / a1) * ( - (a2 * p2))) by A2, RVSUM_1: 23

          .= ((1 / a1) * ((( - 1) * a2) * p2)) by RVSUM_1: 49

          .= ((1 / a1) * (( - a2) * p2));

          

           A4: ((1 / a1) * (a1 * p1)) = ((a1 * (1 / a1)) * p1) by EUCLID_4: 4

          .= (1 * p1) by A2, XCMPLX_1: 106;

          

           A5: (1 * p1) = |[(1 * (p1 . 1)), (1 * (p1 . 2)), (1 * (p1 . 3))]| by Lm1

          .= p1 by Th1;

          

           A6: ( 0.REAL 3) = |[ 0 , 0 , 0 ]| by FINSEQ_2: 62;

          (p1 <X> p2) = (((( - a2) * (1 / a1)) * p2) <X> p2) by A3, A4, A5, EUCLID_4: 4

          .= (((( - a2) / a1) * p2) <X> p2) by XCMPLX_1: 99;

          then (p1 <X> p2) = ((( - a2) / a1) * (p2 <X> p2)) by Lm14;

          

          then (p1 <X> p2) = ((( - a2) / a1) * ( 0.REAL 3)) by FINSEQ_2: 62

          .= |[((( - a2) / a1) * (( 0.REAL 3) . 1)), ((( - a2) / a1) * (( 0.REAL 3) . 2)), ((( - a2) / a1) * (( 0.REAL 3) . 3))]| by Lm1

          .= |[((( - a2) / a1) * 0 ), ((( - a2) / a1) * (( 0.REAL 3) . 2)), ((( - a2) / a1) * (( 0.REAL 3) . 3))]| by A6, FINSEQ_1: 45

          .= |[ 0 , ((( - a2) / a1) * 0 ), ((( - a2) / a1) * (( 0.REAL 3) . 3))]| by A6, FINSEQ_1: 45

          .= ( 0.REAL 3) by A6, FINSEQ_1: 45;

          hence thesis;

        end;

          case ex a1,a2 be Real st ((a1 * p1) + (a2 * p2)) = ( 0.REAL 3) & a2 <> 0 ;

          then

          consider a1,a2 be Real such that

           A7: a2 <> 0 & ((a1 * p1) + (a2 * p2)) = ( 0.REAL 3);

          

           A8: ((1 / a2) * (a2 * p2)) = ((1 / a2) * ( - (a1 * p1))) by A7, RVSUM_1: 23

          .= ((1 / a2) * ((( - 1) * a1) * p1)) by RVSUM_1: 49

          .= ((1 / a2) * (( - a1) * p1));

          

           A9: ((1 / a2) * (a2 * p2)) = ((a2 * (1 / a2)) * p2) by EUCLID_4: 4

          .= (1 * p2) by A7, XCMPLX_1: 106;

          

           A10: (1 * p2) = |[(1 * (p2 . 1)), (1 * (p2 . 2)), (1 * (p2 . 3))]| by Lm1

          .= p2 by Th1;

          

           A11: ( 0.REAL 3) = |[ 0 , 0 , 0 ]| by FINSEQ_2: 62;

          (p1 <X> p2) = (p1 <X> ((( - a1) * (1 / a2)) * p1)) by A8, A9, A10, EUCLID_4: 4

          .= (p1 <X> ((( - a1) / a2) * p1)) by XCMPLX_1: 99

          .= (((( - a1) / a2) * p1) <X> p1) by Lm14

          .= ((( - a1) / a2) * (p1 <X> p1)) by Lm14

          .= ((( - a1) / a2) * ( 0.REAL 3)) by FINSEQ_2: 62

          .= |[((( - a1) / a2) * (( 0.REAL 3) . 1)), ((( - a1) / a2) * (( 0.REAL 3) . 2)), ((( - a1) / a2) * (( 0.REAL 3) . 3))]| by Lm1

          .= |[((( - a1) / a2) * 0 ), ((( - a1) / a2) * (( 0.REAL 3) . 2)), ((( - a1) / a2) * (( 0.REAL 3) . 3))]| by A11, FINSEQ_1: 45

          .= |[ 0 , ((( - a1) / a2) * 0 ), ((( - a1) / a2) * (( 0.REAL 3) . 3))]| by A11, FINSEQ_1: 45

          .= ( 0.REAL 3) by A11, FINSEQ_1: 45;

          hence thesis;

        end;

      end;

      hence thesis;

    end;

    begin

    theorem :: EUCLID_8:4

     |. <e1> .| = 1

    proof

       |( <e1> , <e1> )| = (((1 ^2 ) + ( 0 ^2 )) + ( 0 ^2 )) by Lm12

      .= 1;

      hence thesis by SQUARE_1: 18;

    end;

    theorem :: EUCLID_8:5

     |. <e2> .| = 1

    proof

       |( <e2> , <e2> )| = ((( 0 ^2 ) + (1 ^2 )) + ( 0 ^2 )) by Lm12

      .= 1;

      hence thesis by SQUARE_1: 18;

    end;

    theorem :: EUCLID_8:6

     |. <e3> .| = 1

    proof

       |( <e3> , <e3> )| = ((( 0 ^2 ) + ( 0 ^2 )) + (1 ^2 )) by Lm12

      .= 1;

      hence thesis by SQUARE_1: 18;

    end;

    theorem :: EUCLID_8:7

    ( <e1> , <e2> ) are_orthogonal

    proof

      ( <e1> . 1) = 1 & ( <e1> . 2) = 0 & ( <e1> . 3) = 0 & ( <e2> . 1) = 0 & ( <e2> . 2) = 1 & ( <e2> . 3) = 0 by FINSEQ_1: 45;

      

      then |( <e1> , <e2> )| = (((1 * 0 ) + ( 0 * 1)) + ( 0 * 0 )) by Lm5

      .= 0 ;

      hence ( <e1> , <e2> ) are_orthogonal ;

    end;

    theorem :: EUCLID_8:8

    ( <e1> , <e3> ) are_orthogonal

    proof

      ( <e1> . 1) = 1 & ( <e1> . 2) = 0 & ( <e1> . 3) = 0 & ( <e3> . 1) = 0 & ( <e3> . 2) = 0 & ( <e3> . 3) = 1 by FINSEQ_1: 45;

      

      then |( <e1> , <e3> )| = (((1 * 0 ) + ( 0 * 0 )) + ( 0 * 1)) by Lm5

      .= 0 ;

      hence ( <e1> , <e3> ) are_orthogonal ;

    end;

    theorem :: EUCLID_8:9

    ( <e2> , <e3> ) are_orthogonal

    proof

      ( <e2> . 1) = 0 & ( <e2> . 2) = 1 & ( <e2> . 3) = 0 & ( <e3> . 1) = 0 & ( <e3> . 2) = 0 & ( <e3> . 3) = 1 by FINSEQ_1: 45;

      

      then |( <e2> , <e3> )| = ((( 0 * 0 ) + (1 * 0 )) + ( 0 * 1)) by Lm5

      .= 0 ;

      hence ( <e2> , <e3> ) are_orthogonal ;

    end;

    theorem :: EUCLID_8:10

     |( <e1> , <e1> )| = 1

    proof

      ( <e1> . 1) = 1 & ( <e1> . 2) = 0 & ( <e1> . 3) = 0 by FINSEQ_1: 45;

      

      then |( <e1> , <e1> )| = (((1 * 1) + ( 0 * 0 )) + ( 0 * 0 )) by Lm5

      .= 1;

      hence thesis;

    end;

    theorem :: EUCLID_8:11

     |( <e2> , <e2> )| = 1

    proof

      ( <e2> . 1) = 0 & ( <e2> . 2) = 1 & ( <e2> . 3) = 0 by FINSEQ_1: 45;

      

      then |( <e2> , <e2> )| = ((( 0 * 0 ) + (1 * 1)) + ( 0 * 0 )) by Lm5

      .= 1;

      hence thesis;

    end;

    theorem :: EUCLID_8:12

     |( <e3> , <e3> )| = 1

    proof

      ( <e3> . 1) = 0 & ( <e3> . 2) = 0 & ( <e3> . 3) = 1 by FINSEQ_1: 45;

      

      then |( <e3> , <e3> )| = ((( 0 * 0 ) + ( 0 * 0 )) + (1 * 1)) by Lm5

      .= 1;

      hence thesis;

    end;

    theorem :: EUCLID_8:13

    

     Th13: |(p, |[ 0 , 0 , 0 ]|)| = 0

    proof

      set e = |[ 0 , 0 , 0 ]|;

      (e . 1) = 0 & (e . 2) = 0 & (e . 3) = 0 by FINSEQ_1: 45;

      

      hence |(p, e)| = ((((p . 1) * 0 ) + ((p . 2) * 0 )) + ((p . 3) * 0 )) by Lm5

      .= 0 ;

    end;

    ::$Canceled

    theorem :: EUCLID_8:16

    ( <e1> <X> <e2> ) = <e3>

    proof

      ( <e1> <X> <e2> ) = |[(( 0 * 0 ) - ( 0 * 1)), (( 0 * 0 ) - (1 * 0 )), ((1 * 1) - ( 0 * 0 ))]| by Lm13

      .= <e3> ;

      hence thesis;

    end;

    theorem :: EUCLID_8:17

    ( <e2> <X> <e3> ) = <e1>

    proof

      ( <e2> <X> <e3> ) = |[((1 * 1) - ( 0 * 0 )), (( 0 * 0 ) - ( 0 * 1)), (( 0 * 0 ) - (1 * 0 ))]| by Lm13

      .= <e1> ;

      hence thesis;

    end;

    theorem :: EUCLID_8:18

    ( <e3> <X> <e1> ) = <e2>

    proof

      ( <e3> <X> <e1> ) = |[(( 0 * 0 ) - (1 * 0 )), ((1 * 1) - ( 0 * 0 )), (( 0 * 0 ) - ( 0 * 1))]| by Lm13

      .= <e2> ;

      hence thesis;

    end;

    theorem :: EUCLID_8:19

    ( <e3> <X> <e2> ) = ( - <e1> )

    proof

      

       A1: ( <e1> . 1) = 1 & ( <e1> . 2) = 0 & ( <e1> . 3) = 0 by FINSEQ_1: 45;

      ( <e3> <X> <e2> ) = |[(( 0 * 0 ) - (1 * 1)), ((1 * 0 ) - ( 0 * 0 )), (( 0 * 1) - ( 0 * 0 ))]| by Lm13

      .= |[(( - 1) * ( <e1> . 1)), (( - 1) * ( <e1> . 2)), (( - 1) * ( <e1> . 3))]| by A1

      .= ( - <e1> ) by Lm1;

      hence thesis;

    end;

    theorem :: EUCLID_8:20

    ( <e1> <X> <e3> ) = ( - <e2> )

    proof

      

       A1: ( <e2> . 1) = 0 & ( <e2> . 2) = 1 & ( <e2> . 3) = 0 by FINSEQ_1: 45;

      ( <e1> <X> <e3> ) = |[(( 0 * 1) - ( 0 * 0 )), (( 0 * 0 ) - (1 * 1)), ((1 * 0 ) - ( 0 * 0 ))]| by Lm13

      .= |[(( - 1) * ( <e2> . 1)), (( - 1) * ( <e2> . 2)), (( - 1) * ( <e2> . 3))]| by A1

      .= ( - <e2> ) by Lm1;

      hence thesis;

    end;

    theorem :: EUCLID_8:21

    ( <e2> <X> <e1> ) = ( - <e3> )

    proof

      

       A1: ( <e3> . 1) = 0 & ( <e3> . 2) = 0 & ( <e3> . 3) = 1 by FINSEQ_1: 45;

      ( <e2> <X> <e1> ) = |[((1 * 0 ) - ( 0 * 0 )), (( 0 * 1) - ( 0 * 0 )), (( 0 * 0 ) - (1 * 1))]| by Lm13

      .= |[(( - 1) * ( <e3> . 1)), (( - 1) * ( <e3> . 2)), (( - 1) * ( <e3> . 3))]| by A1

      .= ( - <e3> ) by Lm1;

      hence thesis;

    end;

    theorem :: EUCLID_8:22

    (p <X> |[ 0 , 0 , 0 ]|) = |[ 0 , 0 , 0 ]|

    proof

      p = |[(p . 1), (p . 2), (p . 3)]| by Th1;

      

      hence (p <X> |[ 0 , 0 , 0 ]|) = |[(((p . 2) * 0 ) - ((p . 3) * 0 )), (((p . 3) * 0 ) - ((p . 1) * 0 )), (((p . 1) * 0 ) - ((p . 2) * 0 ))]| by Lm13

      .= |[ 0 , 0 , 0 ]|;

    end;

    ::$Canceled

    theorem :: EUCLID_8:25

    

     Th21: (r * <e1> ) = |[r, 0 , 0 ]|

    proof

      ( <e1> . 1) = 1 & ( <e1> . 2) = 0 & ( <e1> . 3) = 0 by FINSEQ_1: 45;

      

      then (r * <e1> ) = |[(r * 1), (r * 0 ), (r * 0 )]| by Lm1

      .= |[r, 0 , 0 ]|;

      hence thesis;

    end;

    theorem :: EUCLID_8:26

    

     Th22: (r * <e2> ) = |[ 0 , r, 0 ]|

    proof

      ( <e2> . 1) = 0 & ( <e2> . 2) = 1 & ( <e2> . 3) = 0 by FINSEQ_1: 45;

      

      then (r * <e2> ) = |[(r * 0 ), (r * 1), (r * 0 )]| by Lm1

      .= |[ 0 , r, 0 ]|;

      hence thesis;

    end;

    theorem :: EUCLID_8:27

    

     Th23: (r * <e3> ) = |[ 0 , 0 , r]|

    proof

      ( <e3> . 1) = 0 & ( <e3> . 2) = 0 & ( <e3> . 3) = 1 by FINSEQ_1: 45;

      

      then (r * <e3> ) = |[(r * 0 ), (r * 0 ), (r * 1)]| by Lm1

      .= |[ 0 , 0 , r]|;

      hence thesis;

    end;

    theorem :: EUCLID_8:28

    (1 * p) = p by RFUNCT_1: 21;

    ::$Canceled

    theorem :: EUCLID_8:31

    ( - <e1> ) = |[( - 1), 0 , 0 ]|

    proof

      

       A1: ( <e1> . 1) = 1 & ( <e1> . 2) = 0 & ( <e1> . 3) = 0 by FINSEQ_1: 45;

      ( - <e1> ) = |[(( - 1) * ( <e1> . 1)), (( - 1) * ( <e1> . 2)), (( - 1) * ( <e1> . 3))]| by Lm1

      .= |[( - 1), 0 , 0 ]| by A1;

      hence thesis;

    end;

    theorem :: EUCLID_8:32

    ( - <e2> ) = |[ 0 , ( - 1), 0 ]|

    proof

      

       A1: ( <e2> . 1) = 0 & ( <e2> . 2) = 1 & ( <e2> . 3) = 0 by FINSEQ_1: 45;

      ( - <e2> ) = |[(( - 1) * ( <e2> . 1)), (( - 1) * ( <e2> . 2)), (( - 1) * ( <e2> . 3))]| by Lm1

      .= |[ 0 , ( - 1), 0 ]| by A1;

      hence thesis;

    end;

    theorem :: EUCLID_8:33

    ( - <e3> ) = |[ 0 , 0 , ( - 1)]|

    proof

      

       A1: ( <e3> . 1) = 0 & ( <e3> . 2) = 0 & ( <e3> . 3) = 1 by FINSEQ_1: 45;

      ( - <e3> ) = |[(( - 1) * ( <e3> . 1)), (( - 1) * ( <e3> . 2)), (( - 1) * ( <e3> . 3))]| by Lm1

      .= |[ 0 , 0 , ( - 1)]| by A1;

      hence thesis;

    end;

    theorem :: EUCLID_8:34

    ( 0 * p) = |[ 0 , 0 , 0 ]|

    proof

      

      thus ( 0 * p) = |[( 0 * (p . 1)), ( 0 * (p . 2)), ( 0 * (p . 3))]| by Lm1

      .= |[ 0 , 0 , 0 ]|;

    end;

    ::$Canceled

    theorem :: EUCLID_8:37

    p = ((((p . 1) * <e1> ) + ((p . 2) * <e2> )) + ((p . 3) * <e3> ))

    proof

      

       A1: ( <e1> . 1) = 1 & ( <e1> . 2) = 0 & ( <e1> . 3) = 0 by FINSEQ_1: 45;

      

       A2: ( <e2> . 1) = 0 & ( <e2> . 2) = 1 & ( <e2> . 3) = 0 by FINSEQ_1: 45;

      

       A3: ( <e3> . 1) = 0 & ( <e3> . 2) = 0 & ( <e3> . 3) = 1 by FINSEQ_1: 45;

      

       A4: ((((p . 1) * <e1> ) + ((p . 2) * <e2> )) + ((p . 3) * <e3> )) = (( |[((p . 1) * 1), ((p . 1) * 0 ), ((p . 1) * 0 )]| + ((p . 2) * <e2> )) + ((p . 3) * <e3> )) by A1, Lm1

      .= (( |[(p . 1), 0 , 0 ]| + |[((p . 2) * 0 ), ((p . 2) * 1), ((p . 2) * 0 )]|) + ((p . 3) * <e3> )) by A2, Lm1

      .= (( |[(p . 1), 0 , 0 ]| + |[ 0 , (p . 2), 0 ]|) + |[((p . 3) * 0 ), ((p . 3) * 0 ), ((p . 3) * 1)]|) by A3, Lm1

      .= (( |[(p . 1), 0 , 0 ]| + |[ 0 , (p . 2), 0 ]|) + |[ 0 , 0 , (p . 3)]|);

      

       A5: ( |[(p . 1), 0 , 0 ]| . 1) = (p . 1) & ( |[(p . 1), 0 , 0 ]| . 2) = 0 & ( |[(p . 1), 0 , 0 ]| . 3) = 0 by FINSEQ_1: 45;

      

       A6: ( |[ 0 , (p . 2), 0 ]| . 1) = 0 & ( |[ 0 , (p . 2), 0 ]| . 2) = (p . 2) & ( |[ 0 , (p . 2), 0 ]| . 3) = 0 by FINSEQ_1: 45;

      

       A7: ( |[ 0 , 0 , (p . 3)]| . 1) = 0 & ( |[ 0 , 0 , (p . 3)]| . 2) = 0 & ( |[ 0 , 0 , (p . 3)]| . 3) = (p . 3) by FINSEQ_1: 45;

      

       A8: ((((p . 1) * <e1> ) + ((p . 2) * <e2> )) + ((p . 3) * <e3> )) = ( |[((p . 1) + 0 ), ( 0 + (p . 2)), ( 0 + 0 )]| + |[ 0 , 0 , (p . 3)]|) by A4, A5, A6, Lm2

      .= ( |[(p . 1), (p . 2), 0 ]| + |[ 0 , 0 , (p . 3)]|);

      ( |[(p . 1), (p . 2), 0 ]| . 1) = (p . 1) & ( |[(p . 1), (p . 2), 0 ]| . 2) = (p . 2) & ( |[(p . 1), (p . 2), 0 ]| . 3) = 0 by FINSEQ_1: 45;

      

      then ((((p . 1) * <e1> ) + ((p . 2) * <e2> )) + ((p . 3) * <e3> )) = |[((p . 1) + 0 ), ((p . 2) + 0 ), ( 0 + (p . 3))]| by A7, A8, Lm2

      .= |[(p . 1), (p . 2), (p . 3)]|;

      hence p = ((((p . 1) * <e1> ) + ((p . 2) * <e2> )) + ((p . 3) * <e3> )) by Th1;

    end;

    theorem :: EUCLID_8:38

    

     Th30: (r * p) = ((((r * (p . 1)) * <e1> ) + ((r * (p . 2)) * <e2> )) + ((r * (p . 3)) * <e3> ))

    proof

      

       A1: ( <e1> . 1) = 1 & ( <e1> . 2) = 0 & ( <e1> . 3) = 0 by FINSEQ_1: 45;

      

       A2: ( <e2> . 1) = 0 & ( <e2> . 2) = 1 & ( <e2> . 3) = 0 by FINSEQ_1: 45;

      

       A3: ( <e3> . 1) = 0 & ( <e3> . 2) = 0 & ( <e3> . 3) = 1 by FINSEQ_1: 45;

      

       A4: ((((r * (p . 1)) * <e1> ) + ((r * (p . 2)) * <e2> )) + ((r * (p . 3)) * <e3> )) = (( |[((r * (p . 1)) * 1), ((r * (p . 1)) * 0 ), ((r * (p . 1)) * 0 )]| + ((r * (p . 2)) * <e2> )) + ((r * (p . 3)) * <e3> )) by A1, Lm1

      .= (( |[(r * (p . 1)), 0 , 0 ]| + |[((r * (p . 2)) * 0 ), ((r * (p . 2)) * 1), ((r * (p . 2)) * 0 )]|) + ((r * (p . 3)) * <e3> )) by A2, Lm1

      .= (( |[(r * (p . 1)), 0 , 0 ]| + |[ 0 , (r * (p . 2)), 0 ]|) + |[((r * (p . 3)) * 0 ), ((r * (p . 3)) * 0 ), ((r * (p . 3)) * 1)]|) by A3, Lm1

      .= (( |[(r * (p . 1)), 0 , 0 ]| + |[ 0 , (r * (p . 2)), 0 ]|) + |[ 0 , 0 , (r * (p . 3))]|);

      

       A5: ( |[(r * (p . 1)), 0 , 0 ]| . 1) = (r * (p . 1)) & ( |[(r * (p . 1)), 0 , 0 ]| . 2) = 0 & ( |[(r * (p . 1)), 0 , 0 ]| . 3) = 0 by FINSEQ_1: 45;

      

       A6: ( |[ 0 , (r * (p . 2)), 0 ]| . 1) = 0 & ( |[ 0 , (r * (p . 2)), 0 ]| . 2) = (r * (p . 2)) & ( |[ 0 , (r * (p . 2)), 0 ]| . 3) = 0 by FINSEQ_1: 45;

      

       A7: ( |[ 0 , 0 , (r * (p . 3))]| . 1) = 0 & ( |[ 0 , 0 , (r * (p . 3))]| . 2) = 0 & ( |[ 0 , 0 , (r * (p . 3))]| . 3) = (r * (p . 3)) by FINSEQ_1: 45;

      

       A8: ((((r * (p . 1)) * <e1> ) + ((r * (p . 2)) * <e2> )) + ((r * (p . 3)) * <e3> )) = ( |[((r * (p . 1)) + 0 ), ( 0 + (r * (p . 2))), ( 0 + 0 )]| + |[ 0 , 0 , (r * (p . 3))]|) by A4, A5, A6, Lm2

      .= ( |[(r * (p . 1)), (r * (p . 2)), 0 ]| + |[ 0 , 0 , (r * (p . 3))]|);

      ( |[(r * (p . 1)), (r * (p . 2)), 0 ]| . 1) = (r * (p . 1)) & ( |[(r * (p . 1)), (r * (p . 2)), 0 ]| . 2) = (r * (p . 2)) & ( |[(r * (p . 1)), (r * (p . 2)), 0 ]| . 3) = 0 by FINSEQ_1: 45;

      

      then ((((r * (p . 1)) * <e1> ) + ((r * (p . 2)) * <e2> )) + ((r * (p . 3)) * <e3> )) = |[((r * (p . 1)) + 0 ), ((r * (p . 2)) + 0 ), ( 0 + (r * (p . 3)))]| by A7, A8, Lm2

      .= |[(r * (p . 1)), (r * (p . 2)), (r * (p . 3))]|;

      hence (r * p) = ((((r * (p . 1)) * <e1> ) + ((r * (p . 2)) * <e2> )) + ((r * (p . 3)) * <e3> )) by Lm1;

    end;

    theorem :: EUCLID_8:39

    

     Th31: |[x, y, z]| = (((x * <e1> ) + (y * <e2> )) + (z * <e3> ))

    proof

      

       A1: ( <e1> . 1) = 1 & ( <e1> . 2) = 0 & ( <e1> . 3) = 0 by FINSEQ_1: 45;

      

       A2: ( <e2> . 1) = 0 & ( <e2> . 2) = 1 & ( <e2> . 3) = 0 by FINSEQ_1: 45;

      

       A3: ( <e3> . 1) = 0 & ( <e3> . 2) = 0 & ( <e3> . 3) = 1 by FINSEQ_1: 45;

      set p = |[x, y, z]|;

      

       A4: (((x * <e1> ) + (y * <e2> )) + (z * <e3> )) = (( |[(x * 1), (x * 0 ), (x * 0 )]| + (y * <e2> )) + (z * <e3> )) by A1, Lm1

      .= (( |[x, 0 , 0 ]| + |[(y * 0 ), (y * 1), (y * 0 )]|) + (z * <e3> )) by A2, Lm1

      .= (( |[x, 0 , 0 ]| + |[ 0 , y, 0 ]|) + |[(z * 0 ), (z * 0 ), (z * 1)]|) by A3, Lm1

      .= (( |[x, 0 , 0 ]| + |[ 0 , y, 0 ]|) + |[ 0 , 0 , z]|);

      

       A5: ( |[x, 0 , 0 ]| . 1) = x & ( |[x, 0 , 0 ]| . 2) = 0 & ( |[x, 0 , 0 ]| . 3) = 0 by FINSEQ_1: 45;

      

       A6: ( |[ 0 , y, 0 ]| . 1) = 0 & ( |[ 0 , y, 0 ]| . 2) = y & ( |[ 0 , y, 0 ]| . 3) = 0 by FINSEQ_1: 45;

      

       A7: ( |[ 0 , 0 , z]| . 1) = 0 & ( |[ 0 , 0 , z]| . 2) = 0 & ( |[ 0 , 0 , z]| . 3) = z by FINSEQ_1: 45;

      

       A8: (((x * <e1> ) + (y * <e2> )) + (z * <e3> )) = ( |[(x + 0 ), (y + 0 ), ( 0 + 0 )]| + |[ 0 , 0 , z]|) by A4, A5, A6, Lm2

      .= ( |[x, y, 0 ]| + |[ 0 , 0 , z]|);

      ( |[x, y, 0 ]| . 1) = x & ( |[x, y, 0 ]| . 2) = y & ( |[x, y, 0 ]| . 3) = 0 by FINSEQ_1: 45;

      

      then (((x * <e1> ) + (y * <e2> )) + (z * <e3> )) = |[(x + 0 ), (y + 0 ), ( 0 + z)]| by A7, A8, Lm2

      .= |[x, y, z]|;

      hence p = (((x * <e1> ) + (y * <e2> )) + (z * <e3> ));

    end;

    theorem :: EUCLID_8:40

    

     Th32: (r * |[x, y, z]|) = ((((r * x) * <e1> ) + ((r * y) * <e2> )) + ((r * z) * <e3> ))

    proof

      

       A1: ( <e1> . 1) = 1 & ( <e1> . 2) = 0 & ( <e1> . 3) = 0 by FINSEQ_1: 45;

      

       A2: ( <e2> . 1) = 0 & ( <e2> . 2) = 1 & ( <e2> . 3) = 0 by FINSEQ_1: 45;

      

       A3: ( <e3> . 1) = 0 & ( <e3> . 2) = 0 & ( <e3> . 3) = 1 by FINSEQ_1: 45;

      set p = |[x, y, z]|;

      

       A4: ((((r * x) * <e1> ) + ((r * y) * <e2> )) + ((r * z) * <e3> )) = (( |[((r * x) * 1), ((r * x) * 0 ), ((r * x) * 0 )]| + ((r * y) * <e2> )) + ((r * z) * <e3> )) by A1, Lm1

      .= (( |[(r * x), 0 , 0 ]| + |[((r * y) * 0 ), ((r * y) * 1), ((r * y) * 0 )]|) + ((r * z) * <e3> )) by A2, Lm1

      .= (( |[(r * x), 0 , 0 ]| + |[ 0 , (r * y), 0 ]|) + |[((r * z) * 0 ), ((r * z) * 0 ), ((r * z) * 1)]|) by A3, Lm1

      .= (( |[(r * x), 0 , 0 ]| + |[ 0 , (r * y), 0 ]|) + |[ 0 , 0 , (r * z)]|);

      

       A5: ( |[(r * x), 0 , 0 ]| . 1) = (r * x) & ( |[(r * x), 0 , 0 ]| . 2) = 0 & ( |[(r * x), 0 , 0 ]| . 3) = 0 by FINSEQ_1: 45;

      

       A6: ( |[ 0 , (r * y), 0 ]| . 1) = 0 & ( |[ 0 , (r * y), 0 ]| . 2) = (r * y) & ( |[ 0 , (r * y), 0 ]| . 3) = 0 by FINSEQ_1: 45;

      

       A7: ( |[ 0 , 0 , (r * z)]| . 1) = 0 & ( |[ 0 , 0 , (r * z)]| . 2) = 0 & ( |[ 0 , 0 , (r * z)]| . 3) = (r * z) by FINSEQ_1: 45;

      

       A8: ((((r * x) * <e1> ) + ((r * y) * <e2> )) + ((r * z) * <e3> )) = ( |[((r * x) + 0 ), ((r * y) + 0 ), ( 0 + 0 )]| + |[ 0 , 0 , (r * z)]|) by A4, A5, A6, Lm2

      .= ( |[(r * x), (r * y), 0 ]| + |[ 0 , 0 , (r * z)]|);

      ( |[(r * x), (r * y), 0 ]| . 1) = (r * x) & ( |[(r * x), (r * y), 0 ]| . 2) = (r * y) & ( |[(r * x), (r * y), 0 ]| . 3) = 0 by FINSEQ_1: 45;

      

      then ((((r * x) * <e1> ) + ((r * y) * <e2> )) + ((r * z) * <e3> )) = |[((r * x) + 0 ), ((r * y) + 0 ), ( 0 + (r * z))]| by A7, A8, Lm2

      .= |[(r * x), (r * y), (r * z)]|;

      hence (r * p) = ((((r * x) * <e1> ) + ((r * y) * <e2> )) + ((r * z) * <e3> )) by Lm6;

    end;

    theorem :: EUCLID_8:41

    ( - p) = ((( - ((p . 1) * <e1> )) - ((p . 2) * <e2> )) - ((p . 3) * <e3> ))

    proof

      ( - p) = ((((( - 1) * (p . 1)) * <e1> ) + ((( - 1) * (p . 2)) * <e2> )) + ((( - 1) * (p . 3)) * <e3> )) by Th30

      .= ((( - ((p . 1) * <e1> )) + ((( - 1) * (p . 2)) * <e2> )) + (( - (p . 3)) * <e3> )) by RVSUM_1: 49

      .= ((( - ((p . 1) * <e1> )) + ( - ((p . 2) * <e2> ))) + ((( - 1) * (p . 3)) * <e3> )) by RVSUM_1: 49

      .= ((( - ((p . 1) * <e1> )) - ((p . 2) * <e2> )) - ((p . 3) * <e3> )) by RVSUM_1: 49;

      hence thesis;

    end;

    theorem :: EUCLID_8:42

    ( - |[x, y, z]|) = ((( - (x * <e1> )) - (y * <e2> )) - (z * <e3> ))

    proof

      ( - |[x, y, z]|) = ((((( - 1) * x) * <e1> ) + ((( - 1) * y) * <e2> )) + ((( - 1) * z) * <e3> )) by Th32

      .= ((( - (x * <e1> )) + ((( - 1) * y) * <e2> )) + (( - z) * <e3> )) by RVSUM_1: 49

      .= ((( - (x * <e1> )) + ( - (y * <e2> ))) + ((( - 1) * z) * <e3> )) by RVSUM_1: 49

      .= ((( - (x * <e1> )) - (y * <e2> )) - (z * <e3> )) by RVSUM_1: 49;

      hence thesis;

    end;

    theorem :: EUCLID_8:43

    (p1 + p2) = (((((p1 . 1) + (p2 . 1)) * <e1> ) + (((p1 . 2) + (p2 . 2)) * <e2> )) + (((p1 . 3) + (p2 . 3)) * <e3> ))

    proof

      

       A1: ( <e1> . 1) = 1 & ( <e1> . 2) = 0 & ( <e1> . 3) = 0 by FINSEQ_1: 45;

      

       A2: ( <e2> . 1) = 0 & ( <e2> . 2) = 1 & ( <e2> . 3) = 0 by FINSEQ_1: 45;

      

       A3: ( <e3> . 1) = 0 & ( <e3> . 2) = 0 & ( <e3> . 3) = 1 by FINSEQ_1: 45;

      

       A4: ((p1 + p2) . 1) = ((p1 . 1) + (p2 . 1)) by RVSUM_1: 11;

      

       A5: ((p1 + p2) . 2) = ((p1 . 2) + (p2 . 2)) by RVSUM_1: 11;

      

       A6: ((p1 + p2) . 3) = ((p1 . 3) + (p2 . 3)) by RVSUM_1: 11;

      

       A7: (((((p1 + p2) . 1) * <e1> ) + (((p1 + p2) . 2) * <e2> )) + (((p1 + p2) . 3) * <e3> )) = (( |[(((p1 + p2) . 1) * 1), (((p1 + p2) . 1) * 0 ), (((p1 + p2) . 1) * 0 )]| + (((p1 + p2) . 2) * <e2> )) + (((p1 + p2) . 3) * <e3> )) by A1, Lm1

      .= (( |[((p1 + p2) . 1), 0 , 0 ]| + |[(((p1 + p2) . 2) * 0 ), (((p1 + p2) . 2) * 1), (((p1 + p2) . 2) * 0 )]|) + (((p1 + p2) . 3) * <e3> )) by A2, Lm1

      .= (( |[((p1 + p2) . 1), 0 , 0 ]| + |[ 0 , ((p1 + p2) . 2), 0 ]|) + |[(((p1 + p2) . 3) * 0 ), (((p1 + p2) . 3) * 0 ), (((p1 + p2) . 3) * 1)]|) by A3, Lm1

      .= (( |[((p1 + p2) . 1), 0 , 0 ]| + |[ 0 , ((p1 + p2) . 2), 0 ]|) + |[ 0 , 0 , ((p1 + p2) . 3)]|);

      

       A8: ( |[((p1 + p2) . 1), 0 , 0 ]| . 1) = ((p1 + p2) . 1) & ( |[((p1 + p2) . 1), 0 , 0 ]| . 2) = 0 & ( |[((p1 + p2) . 1), 0 , 0 ]| . 3) = 0 by FINSEQ_1: 45;

      

       A9: ( |[ 0 , ((p1 + p2) . 2), 0 ]| . 1) = 0 & ( |[ 0 , ((p1 + p2) . 2), 0 ]| . 2) = ((p1 + p2) . 2) & ( |[ 0 , ((p1 + p2) . 2), 0 ]| . 3) = 0 by FINSEQ_1: 45;

      

       A10: ( |[ 0 , 0 , ((p1 + p2) . 3)]| . 1) = 0 & ( |[ 0 , 0 , ((p1 + p2) . 3)]| . 2) = 0 & ( |[ 0 , 0 , ((p1 + p2) . 3)]| . 3) = ((p1 + p2) . 3) by FINSEQ_1: 45;

      

       A11: (((((p1 + p2) . 1) * <e1> ) + (((p1 + p2) . 2) * <e2> )) + (((p1 + p2) . 3) * <e3> )) = ( |[(((p1 + p2) . 1) + 0 ), ( 0 + ((p1 + p2) . 2)), ( 0 + 0 )]| + |[ 0 , 0 , ((p1 + p2) . 3)]|) by A7, A8, A9, Lm2

      .= ( |[((p1 + p2) . 1), ((p1 + p2) . 2), 0 ]| + |[ 0 , 0 , ((p1 + p2) . 3)]|);

      ( |[((p1 + p2) . 1), ((p1 + p2) . 2), 0 ]| . 1) = ((p1 + p2) . 1) & ( |[((p1 + p2) . 1), ((p1 + p2) . 2), 0 ]| . 2) = ((p1 + p2) . 2) & ( |[((p1 + p2) . 1), ((p1 + p2) . 2), 0 ]| . 3) = 0 by FINSEQ_1: 45;

      

      then (((((p1 + p2) . 1) * <e1> ) + (((p1 + p2) . 2) * <e2> )) + (((p1 + p2) . 3) * <e3> )) = |[(((p1 + p2) . 1) + 0 ), (((p1 + p2) . 2) + 0 ), ( 0 + ((p1 + p2) . 3))]| by A10, A11, Lm2

      .= |[((p1 + p2) . 1), ((p1 + p2) . 2), ((p1 + p2) . 3)]|;

      hence thesis by A4, A5, A6, Th1;

    end;

    theorem :: EUCLID_8:44

    

     Th36: (p1 - p2) = (((((p1 . 1) - (p2 . 1)) * <e1> ) + (((p1 . 2) - (p2 . 2)) * <e2> )) + (((p1 . 3) - (p2 . 3)) * <e3> ))

    proof

      

       A1: ( <e1> . 1) = 1 & ( <e1> . 2) = 0 & ( <e1> . 3) = 0 by FINSEQ_1: 45;

      

       A2: ( <e2> . 1) = 0 & ( <e2> . 2) = 1 & ( <e2> . 3) = 0 by FINSEQ_1: 45;

      

       A3: ( <e3> . 1) = 0 & ( <e3> . 2) = 0 & ( <e3> . 3) = 1 by FINSEQ_1: 45;

      

       A4: ((p1 - p2) . 1) = ((p1 . 1) - (p2 . 1)) by RVSUM_1: 27;

      

       A5: ((p1 - p2) . 2) = ((p1 . 2) - (p2 . 2)) by RVSUM_1: 27;

      

       A6: ((p1 - p2) . 3) = ((p1 . 3) - (p2 . 3)) by RVSUM_1: 27;

      

       A7: (((((p1 - p2) . 1) * <e1> ) + (((p1 - p2) . 2) * <e2> )) + (((p1 - p2) . 3) * <e3> )) = (( |[(((p1 - p2) . 1) * 1), (((p1 - p2) . 1) * 0 ), (((p1 - p2) . 1) * 0 )]| + (((p1 - p2) . 2) * <e2> )) + (((p1 - p2) . 3) * <e3> )) by A1, Lm1

      .= (( |[((p1 - p2) . 1), 0 , 0 ]| + |[(((p1 - p2) . 2) * 0 ), (((p1 - p2) . 2) * 1), (((p1 - p2) . 2) * 0 )]|) + (((p1 - p2) . 3) * <e3> )) by A2, Lm1

      .= (( |[((p1 - p2) . 1), 0 , 0 ]| + |[ 0 , ((p1 - p2) . 2), 0 ]|) + |[(((p1 - p2) . 3) * 0 ), (((p1 - p2) . 3) * 0 ), (((p1 - p2) . 3) * 1)]|) by A3, Lm1

      .= (( |[((p1 - p2) . 1), 0 , 0 ]| + |[ 0 , ((p1 - p2) . 2), 0 ]|) + |[ 0 , 0 , ((p1 - p2) . 3)]|);

      

       A8: ( |[((p1 - p2) . 1), 0 , 0 ]| . 1) = ((p1 - p2) . 1) & ( |[((p1 - p2) . 1), 0 , 0 ]| . 2) = 0 & ( |[((p1 - p2) . 1), 0 , 0 ]| . 3) = 0 by FINSEQ_1: 45;

      

       A9: ( |[ 0 , ((p1 - p2) . 2), 0 ]| . 1) = 0 & ( |[ 0 , ((p1 - p2) . 2), 0 ]| . 2) = ((p1 - p2) . 2) & ( |[ 0 , ((p1 - p2) . 2), 0 ]| . 3) = 0 by FINSEQ_1: 45;

      

       A10: ( |[ 0 , 0 , ((p1 - p2) . 3)]| . 1) = 0 & ( |[ 0 , 0 , ((p1 - p2) . 3)]| . 2) = 0 & ( |[ 0 , 0 , ((p1 - p2) . 3)]| . 3) = ((p1 - p2) . 3) by FINSEQ_1: 45;

      

       A11: (((((p1 - p2) . 1) * <e1> ) + (((p1 - p2) . 2) * <e2> )) + (((p1 - p2) . 3) * <e3> )) = ( |[(((p1 - p2) . 1) + 0 ), ( 0 + ((p1 - p2) . 2)), ( 0 + 0 )]| + |[ 0 , 0 , ((p1 - p2) . 3)]|) by A7, A8, A9, Lm2

      .= ( |[((p1 - p2) . 1), ((p1 - p2) . 2), 0 ]| + |[ 0 , 0 , ((p1 - p2) . 3)]|);

      ( |[((p1 - p2) . 1), ((p1 - p2) . 2), 0 ]| . 1) = ((p1 - p2) . 1) & ( |[((p1 - p2) . 1), ((p1 - p2) . 2), 0 ]| . 2) = ((p1 - p2) . 2) & ( |[((p1 - p2) . 1), ((p1 - p2) . 2), 0 ]| . 3) = 0 by FINSEQ_1: 45;

      

      then (((((p1 - p2) . 1) * <e1> ) + (((p1 - p2) . 2) * <e2> )) + (((p1 - p2) . 3) * <e3> )) = |[(((p1 - p2) . 1) + 0 ), (((p1 - p2) . 2) + 0 ), ( 0 + ((p1 - p2) . 3))]| by A10, A11, Lm2

      .= |[((p1 - p2) . 1), ((p1 - p2) . 2), ((p1 - p2) . 3)]|;

      hence thesis by A4, A5, A6, Th1;

    end;

    theorem :: EUCLID_8:45

    ( |[x1, x2, x3]| + |[y1, y2, y3]|) = ((((x1 + y1) * <e1> ) + ((x2 + y2) * <e2> )) + ((x3 + y3) * <e3> ))

    proof

      

       A1: ( <e1> . 1) = 1 & ( <e1> . 2) = 0 & ( <e1> . 3) = 0 by FINSEQ_1: 45;

      

       A2: ( <e2> . 1) = 0 & ( <e2> . 2) = 1 & ( <e2> . 3) = 0 by FINSEQ_1: 45;

      

       A3: ( <e3> . 1) = 0 & ( <e3> . 2) = 0 & ( <e3> . 3) = 1 by FINSEQ_1: 45;

      

       A4: ((((x1 + y1) * <e1> ) + ((x2 + y2) * <e2> )) + ((x3 + y3) * <e3> )) = (( |[((x1 + y1) * 1), ((x1 + y1) * 0 ), ((x1 + y1) * 0 )]| + ((x2 + y2) * <e2> )) + ((x3 + y3) * <e3> )) by A1, Lm1

      .= (( |[(x1 + y1), 0 , 0 ]| + |[((x2 + y2) * 0 ), ((x2 + y2) * 1), ((x2 + y2) * 0 )]|) + ((x3 + y3) * <e3> )) by A2, Lm1

      .= (( |[(x1 + y1), 0 , 0 ]| + |[ 0 , (x2 + y2), 0 ]|) + |[((x3 + y3) * 0 ), ((x3 + y3) * 0 ), ((x3 + y3) * 1)]|) by A3, Lm1

      .= (( |[(x1 + y1), 0 , 0 ]| + |[ 0 , (x2 + y2), 0 ]|) + |[ 0 , 0 , (x3 + y3)]|);

      

       A5: ( |[(x1 + y1), 0 , 0 ]| . 1) = (x1 + y1) & ( |[(x1 + y1), 0 , 0 ]| . 2) = 0 & ( |[(x1 + y1), 0 , 0 ]| . 3) = 0 by FINSEQ_1: 45;

      

       A6: ( |[ 0 , (x2 + y2), 0 ]| . 1) = 0 & ( |[ 0 , (x2 + y2), 0 ]| . 2) = (x2 + y2) & ( |[ 0 , (x2 + y2), 0 ]| . 3) = 0 by FINSEQ_1: 45;

      

       A7: ( |[ 0 , 0 , (x3 + y3)]| . 1) = 0 & ( |[ 0 , 0 , (x3 + y3)]| . 2) = 0 & ( |[ 0 , 0 , (x3 + y3)]| . 3) = (x3 + y3) by FINSEQ_1: 45;

      

       A8: ((((x1 + y1) * <e1> ) + ((x2 + y2) * <e2> )) + ((x3 + y3) * <e3> )) = ( |[((x1 + y1) + 0 ), ( 0 + (x2 + y2)), ( 0 + 0 )]| + |[ 0 , 0 , (x3 + y3)]|) by A4, A5, A6, Lm2

      .= ( |[(x1 + y1), (x2 + y2), 0 ]| + |[ 0 , 0 , (x3 + y3)]|);

      ( |[(x1 + y1), (x2 + y2), 0 ]| . 1) = (x1 + y1) & ( |[(x1 + y1), (x2 + y2), 0 ]| . 2) = (x2 + y2) & ( |[(x1 + y1), (x2 + y2), 0 ]| . 3) = 0 by FINSEQ_1: 45;

      

      then

       A9: ((((x1 + y1) * <e1> ) + ((x2 + y2) * <e2> )) + ((x3 + y3) * <e3> )) = |[((x1 + y1) + 0 ), ((x2 + y2) + 0 ), ( 0 + (x3 + y3))]| by A8, A7, Lm2

      .= |[(x1 + y1), (x2 + y2), (x3 + y3)]|;

      

       A10: ( |[x1, x2, x3]| . 1) = x1 & ( |[x1, x2, x3]| . 2) = x2 & ( |[x1, x2, x3]| . 3) = x3 by FINSEQ_1: 45;

      ( |[y1, y2, y3]| . 1) = y1 & ( |[y1, y2, y3]| . 2) = y2 & ( |[y1, y2, y3]| . 3) = y3 by FINSEQ_1: 45;

      hence thesis by A10, Lm2, A9;

    end;

    theorem :: EUCLID_8:46

    ( |[x1, x2, x3]| - |[y1, y2, y3]|) = ((((x1 - y1) * <e1> ) + ((x2 - y2) * <e2> )) + ((x3 - y3) * <e3> ))

    proof

      

       A1: ( |[y1, y2, y3]| . 1) = y1 by FINSEQ_1: 45;

      

       A2: ( |[y1, y2, y3]| . 2) = y2 by FINSEQ_1: 45;

      

       A3: ( |[y1, y2, y3]| . 3) = y3 by FINSEQ_1: 45;

      

       A4: (( |[x1, x2, x3]| . 1) - ( |[y1, y2, y3]| . 1)) = (x1 - y1) by A1, FINSEQ_1: 45;

      

       A5: (( |[x1, x2, x3]| . 2) - ( |[y1, y2, y3]| . 2)) = (x2 - y2) by A2, FINSEQ_1: 45;

      (( |[x1, x2, x3]| . 3) - ( |[y1, y2, y3]| . 3)) = (x3 - y3) by A3, FINSEQ_1: 45;

      hence thesis by A4, A5, Th36;

    end;

    theorem :: EUCLID_8:47

    ((((p1 . 1) * <e1> ) + ((p1 . 2) * <e2> )) + ((p1 . 3) * <e3> )) = (((((p2 . 1) + (p3 . 1)) * <e1> ) + (((p2 . 2) + (p3 . 2)) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) iff ((((p2 . 1) * <e1> ) + ((p2 . 2) * <e2> )) + ((p2 . 3) * <e3> )) = (((((p1 . 1) - (p3 . 1)) * <e1> ) + (((p1 . 2) - (p3 . 2)) * <e2> )) + (((p1 . 3) - (p3 . 3)) * <e3> ))

    proof

      

       A1: ((r1 * R) - (r2 * R)) = ((r1 - r2) * R)

      proof

        ((r1 * R) - (r2 * R)) = ((r1 * R) + ((( - 1) * r2) * R)) by RVSUM_1: 49

        .= ((r1 + ( - r2)) * R) by RVSUM_1: 50;

        hence thesis;

      end;

      

       A2: ( <e1> . 1) = 1 & ( <e1> . 2) = 0 & ( <e1> . 3) = 0 by FINSEQ_1: 45;

      

       A3: ( <e2> . 1) = 0 & ( <e2> . 2) = 1 & ( <e2> . 3) = 0 by FINSEQ_1: 45;

      

       A4: ( <e3> . 1) = 0 & ( <e3> . 2) = 0 & ( <e3> . 3) = 1 by FINSEQ_1: 45;

      thus ((((p1 . 1) * <e1> ) + ((p1 . 2) * <e2> )) + ((p1 . 3) * <e3> )) = (((((p2 . 1) + (p3 . 1)) * <e1> ) + (((p2 . 2) + (p3 . 2)) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) implies ((((p2 . 1) * <e1> ) + ((p2 . 2) * <e2> )) + ((p2 . 3) * <e3> )) = (((((p1 . 1) - (p3 . 1)) * <e1> ) + (((p1 . 2) - (p3 . 2)) * <e2> )) + (((p1 . 3) - (p3 . 3)) * <e3> ))

      proof

        assume ((((p1 . 1) * <e1> ) + ((p1 . 2) * <e2> )) + ((p1 . 3) * <e3> )) = (((((p2 . 1) + (p3 . 1)) * <e1> ) + (((p2 . 2) + (p3 . 2)) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> ));

        then ((((((p1 . 1) * <e1> ) + (((p1 . 2) * <e2> ) + ((p1 . 3) * <e3> ))) - ((p3 . 1) * <e1> )) - ((p3 . 2) * <e2> )) - ((p3 . 3) * <e3> )) = ((((((((p2 . 1) + (p3 . 1)) * <e1> ) + (((p2 . 2) + (p3 . 2)) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) - ((p3 . 1) * <e1> )) - ((p3 . 2) * <e2> )) - ((p3 . 3) * <e3> )) by RVSUM_1: 15;

        then ((((((p1 . 1) * <e1> ) - ((p3 . 1) * <e1> )) + (((p1 . 2) * <e2> ) + ((p1 . 3) * <e3> ))) - ((p3 . 2) * <e2> )) - ((p3 . 3) * <e3> )) = ((((((((p2 . 1) + (p3 . 1)) * <e1> ) + (((p2 . 2) + (p3 . 2)) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) - ((p3 . 1) * <e1> )) - ((p3 . 2) * <e2> )) - ((p3 . 3) * <e3> )) by RVSUM_1: 15;

        then ((((((p1 . 1) - (p3 . 1)) * <e1> ) + (((p1 . 2) * <e2> ) + ((p1 . 3) * <e3> ))) - ((p3 . 2) * <e2> )) - ((p3 . 3) * <e3> )) = ((((((((p2 . 1) + (p3 . 1)) * <e1> ) + (((p2 . 2) + (p3 . 2)) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) - ((p3 . 1) * <e1> )) - ((p3 . 2) * <e2> )) - ((p3 . 3) * <e3> )) by A1;

        then (((((((p1 . 1) - (p3 . 1)) * <e1> ) + ((p1 . 2) * <e2> )) + ((p1 . 3) * <e3> )) - ((p3 . 2) * <e2> )) - ((p3 . 3) * <e3> )) = ((((((((p2 . 1) + (p3 . 1)) * <e1> ) + (((p2 . 2) + (p3 . 2)) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) - ((p3 . 1) * <e1> )) - ((p3 . 2) * <e2> )) - ((p3 . 3) * <e3> )) by RVSUM_1: 15;

        then (((((((p1 . 1) - (p3 . 1)) * <e1> ) + ((p1 . 2) * <e2> )) - ((p3 . 2) * <e2> )) + ((p1 . 3) * <e3> )) - ((p3 . 3) * <e3> )) = ((((((((p2 . 1) + (p3 . 1)) * <e1> ) + (((p2 . 2) + (p3 . 2)) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) - ((p3 . 1) * <e1> )) - ((p3 . 2) * <e2> )) - ((p3 . 3) * <e3> )) by RVSUM_1: 15;

        then ((((((p1 . 1) - (p3 . 1)) * <e1> ) + (((p1 . 2) * <e2> ) + ( - ((p3 . 2) * <e2> )))) + ((p1 . 3) * <e3> )) + ( - ((p3 . 3) * <e3> ))) = ((((((((p2 . 1) + (p3 . 1)) * <e1> ) + (((p2 . 2) + (p3 . 2)) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) - ((p3 . 1) * <e1> )) - ((p3 . 2) * <e2> )) - ((p3 . 3) * <e3> )) by RVSUM_1: 15;

        then (((((p1 . 1) - (p3 . 1)) * <e1> ) + (((p1 . 2) * <e2> ) - ((p3 . 2) * <e2> ))) + (((p1 . 3) * <e3> ) - ((p3 . 3) * <e3> ))) = ((((((((p2 . 1) + (p3 . 1)) * <e1> ) + (((p2 . 2) + (p3 . 2)) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) - ((p3 . 1) * <e1> )) - ((p3 . 2) * <e2> )) - ((p3 . 3) * <e3> )) by RVSUM_1: 15;

        then (((((p1 . 1) - (p3 . 1)) * <e1> ) + (((p1 . 2) - (p3 . 2)) * <e2> )) + (((p1 . 3) * <e3> ) - ((p3 . 3) * <e3> ))) = ((((((((p2 . 1) + (p3 . 1)) * <e1> ) + (((p2 . 2) + (p3 . 2)) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) - ((p3 . 1) * <e1> )) - ((p3 . 2) * <e2> )) - ((p3 . 3) * <e3> )) by A1;

        then (((((p1 . 1) - (p3 . 1)) * <e1> ) + (((p1 . 2) - (p3 . 2)) * <e2> )) + (((p1 . 3) - (p3 . 3)) * <e3> )) = ((((((((p2 . 1) + (p3 . 1)) * <e1> ) + (((p2 . 2) + (p3 . 2)) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) - ((p3 . 1) * <e1> )) - ((p3 . 2) * <e2> )) - ((p3 . 3) * <e3> )) by A1;

        then (((((p1 . 1) - (p3 . 1)) * <e1> ) + (((p1 . 2) - (p3 . 2)) * <e2> )) + (((p1 . 3) - (p3 . 3)) * <e3> )) = ((((((((p2 . 1) + (p3 . 1)) * <e1> ) + (((p2 . 2) + (p3 . 2)) * <e2> )) - ((p3 . 1) * <e1> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) - ((p3 . 2) * <e2> )) - ((p3 . 3) * <e3> )) by RVSUM_1: 15;

        then (((((p1 . 1) - (p3 . 1)) * <e1> ) + (((p1 . 2) - (p3 . 2)) * <e2> )) + (((p1 . 3) - (p3 . 3)) * <e3> )) = ((((((((p2 . 1) + (p3 . 1)) * <e1> ) - ((p3 . 1) * <e1> )) + (((p2 . 2) + (p3 . 2)) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) - ((p3 . 2) * <e2> )) - ((p3 . 3) * <e3> )) by RVSUM_1: 15;

        then (((((p1 . 1) - (p3 . 1)) * <e1> ) + (((p1 . 2) - (p3 . 2)) * <e2> )) + (((p1 . 3) - (p3 . 3)) * <e3> )) = ((((((((p2 . 1) + (p3 . 1)) * <e1> ) - ((p3 . 1) * <e1> )) + (((p2 . 2) + (p3 . 2)) * <e2> )) - ((p3 . 2) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) - ((p3 . 3) * <e3> )) by RVSUM_1: 15;

        then (((((p1 . 1) - (p3 . 1)) * <e1> ) + (((p1 . 2) - (p3 . 2)) * <e2> )) + (((p1 . 3) - (p3 . 3)) * <e3> )) = ((((((((p2 . 1) * <e1> ) + ((p3 . 1) * <e1> )) + ( - ((p3 . 1) * <e1> ))) + (((p2 . 2) + (p3 . 2)) * <e2> )) - ((p3 . 2) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) - ((p3 . 3) * <e3> )) by RVSUM_1: 50;

        then (((((p1 . 1) - (p3 . 1)) * <e1> ) + (((p1 . 2) - (p3 . 2)) * <e2> )) + (((p1 . 3) - (p3 . 3)) * <e3> )) = (((((((p2 . 1) * <e1> ) + (((p3 . 1) * <e1> ) - ((p3 . 1) * <e1> ))) + (((p2 . 2) + (p3 . 2)) * <e2> )) - ((p3 . 2) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) - ((p3 . 3) * <e3> )) by RVSUM_1: 15;

        then (((((p1 . 1) - (p3 . 1)) * <e1> ) + (((p1 . 2) - (p3 . 2)) * <e2> )) + (((p1 . 3) - (p3 . 3)) * <e3> )) = (((((((p2 . 1) * <e1> ) + ( 0.REAL 3)) + (((p2 . 2) + (p3 . 2)) * <e2> )) - ((p3 . 2) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) - ((p3 . 3) * <e3> )) by EUCLIDLP: 2;

        then (((((p1 . 1) - (p3 . 1)) * <e1> ) + (((p1 . 2) - (p3 . 2)) * <e2> )) + (((p1 . 3) - (p3 . 3)) * <e3> )) = ((((((p2 . 1) * <e1> ) + (((p2 . 2) + (p3 . 2)) * <e2> )) - ((p3 . 2) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) - ((p3 . 3) * <e3> )) by EUCLID_4: 1;

        then (((((p1 . 1) - (p3 . 1)) * <e1> ) + (((p1 . 2) - (p3 . 2)) * <e2> )) + (((p1 . 3) - (p3 . 3)) * <e3> )) = ((((((p2 . 1) * <e1> ) + (((p2 . 2) * <e2> ) + ((p3 . 2) * <e2> ))) - ((p3 . 2) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) - ((p3 . 3) * <e3> )) by RVSUM_1: 50;

        then (((((p1 . 1) - (p3 . 1)) * <e1> ) + (((p1 . 2) - (p3 . 2)) * <e2> )) + (((p1 . 3) - (p3 . 3)) * <e3> )) = (((((((p2 . 1) * <e1> ) + ((p2 . 2) * <e2> )) + ((p3 . 2) * <e2> )) + ( - ((p3 . 2) * <e2> ))) + (((p2 . 3) + (p3 . 3)) * <e3> )) - ((p3 . 3) * <e3> )) by RVSUM_1: 15;

        then (((((p1 . 1) - (p3 . 1)) * <e1> ) + (((p1 . 2) - (p3 . 2)) * <e2> )) + (((p1 . 3) - (p3 . 3)) * <e3> )) = ((((((p2 . 1) * <e1> ) + ((p2 . 2) * <e2> )) + (((p3 . 2) * <e2> ) + ( - ((p3 . 2) * <e2> )))) + (((p2 . 3) + (p3 . 3)) * <e3> )) - ((p3 . 3) * <e3> )) by RVSUM_1: 15;

        then (((((p1 . 1) - (p3 . 1)) * <e1> ) + (((p1 . 2) - (p3 . 2)) * <e2> )) + (((p1 . 3) - (p3 . 3)) * <e3> )) = ((((((p2 . 1) * <e1> ) + ((p2 . 2) * <e2> )) + ( 0.REAL 3)) + (((p2 . 3) + (p3 . 3)) * <e3> )) - ((p3 . 3) * <e3> )) by EUCLIDLP: 2;

        then (((((p1 . 1) - (p3 . 1)) * <e1> ) + (((p1 . 2) - (p3 . 2)) * <e2> )) + (((p1 . 3) - (p3 . 3)) * <e3> )) = (((((p2 . 1) * <e1> ) + ((p2 . 2) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) - ((p3 . 3) * <e3> )) by EUCLID_4: 1;

        then (((((p1 . 1) - (p3 . 1)) * <e1> ) + (((p1 . 2) - (p3 . 2)) * <e2> )) + (((p1 . 3) - (p3 . 3)) * <e3> )) = (((((p2 . 1) * <e1> ) + ((p2 . 2) * <e2> )) + (((p2 . 3) * <e3> ) + ((p3 . 3) * <e3> ))) - ((p3 . 3) * <e3> )) by RVSUM_1: 50;

        then (((((p1 . 1) - (p3 . 1)) * <e1> ) + (((p1 . 2) - (p3 . 2)) * <e2> )) + (((p1 . 3) - (p3 . 3)) * <e3> )) = ((((((p2 . 1) * <e1> ) + ((p2 . 2) * <e2> )) + ((p2 . 3) * <e3> )) + ((p3 . 3) * <e3> )) - ((p3 . 3) * <e3> )) by RVSUM_1: 15;

        then (((((p1 . 1) - (p3 . 1)) * <e1> ) + (((p1 . 2) - (p3 . 2)) * <e2> )) + (((p1 . 3) - (p3 . 3)) * <e3> )) = (((((p2 . 1) * <e1> ) + ((p2 . 2) * <e2> )) + ((p2 . 3) * <e3> )) + (((p3 . 3) * <e3> ) + ( - ((p3 . 3) * <e3> )))) by RVSUM_1: 15;

        then (((((p1 . 1) - (p3 . 1)) * <e1> ) + (((p1 . 2) - (p3 . 2)) * <e2> )) + (((p1 . 3) - (p3 . 3)) * <e3> )) = (((((p2 . 1) * <e1> ) + ((p2 . 2) * <e2> )) + ((p2 . 3) * <e3> )) + ( 0.REAL 3)) by EUCLIDLP: 2;

        hence thesis by EUCLID_4: 1;

      end;

      now

        assume

         A5: ((((p2 . 1) * <e1> ) + ((p2 . 2) * <e2> )) + ((p2 . 3) * <e3> )) = (((((p1 . 1) - (p3 . 1)) * <e1> ) + (((p1 . 2) - (p3 . 2)) * <e2> )) + (((p1 . 3) - (p3 . 3)) * <e3> ));

        (((((((p2 . 1) * <e1> ) + ((p2 . 2) * <e2> )) + ((p2 . 3) * <e3> )) + ((p3 . 1) * <e1> )) + ((p3 . 2) * <e2> )) + ((p3 . 3) * <e3> )) = (((((p2 . 1) * <e1> ) + ((p2 . 2) * <e2> )) + ((p2 . 3) * <e3> )) + ((((p3 . 1) * <e1> ) + ((p3 . 2) * <e2> )) + ((p3 . 3) * <e3> )))

        proof

          

           A6: (((p2 . 1) * <e1> ) . 1) = ((p2 . 1) * 1) by A2, RVSUM_1: 44

          .= (p2 . 1);

          

           A7: (((p2 . 1) * <e1> ) . 2) = ((p2 . 1) * 0 ) by A2, RVSUM_1: 44

          .= 0 ;

          

           A8: (((p2 . 1) * <e1> ) . 3) = ((p2 . 1) * 0 ) by A2, RVSUM_1: 44

          .= 0 ;

          

           A9: (((p2 . 2) * <e2> ) . 1) = ((p2 . 2) * ( <e2> . 1)) by RVSUM_1: 44

          .= 0 by A3;

          

           A10: (((p2 . 2) * <e2> ) . 2) = ((p2 . 2) * 1) by A3, RVSUM_1: 44

          .= (p2 . 2);

          

           A11: (((p2 . 2) * <e2> ) . 3) = ((p2 . 2) * 0 ) by A3, RVSUM_1: 44

          .= 0 ;

          

           A12: (((p2 . 3) * <e3> ) . 1) = ((p2 . 3) * 0 ) by A4, RVSUM_1: 44

          .= 0 ;

          

           A13: (((p2 . 3) * <e3> ) . 2) = ((p2 . 3) * 0 ) by A4, RVSUM_1: 44

          .= 0 ;

          

           A14: (((p2 . 3) * <e3> ) . 3) = ((p2 . 3) * 1) by A4, RVSUM_1: 44

          .= (p2 . 3);

          

           A15: (((p3 . 1) * <e1> ) . 1) = ((p3 . 1) * 1) by A2, RVSUM_1: 44

          .= (p3 . 1);

          

           A16: (((p3 . 1) * <e1> ) . 2) = ((p3 . 1) * 0 ) by A2, RVSUM_1: 44

          .= 0 ;

          

           A17: (((p3 . 1) * <e1> ) . 3) = ((p3 . 1) * 0 ) by A2, RVSUM_1: 44

          .= 0 ;

          

           A18: (((p3 . 2) * <e2> ) . 1) = ((p3 . 2) * ( <e2> . 1)) by RVSUM_1: 44

          .= 0 by A3;

          

           A19: (((p3 . 2) * <e2> ) . 2) = ((p3 . 2) * 1) by A3, RVSUM_1: 44

          .= (p3 . 2);

          

           A20: (((p3 . 2) * <e2> ) . 3) = ((p3 . 2) * 0 ) by A3, RVSUM_1: 44

          .= 0 ;

          

           A21: (((p3 . 3) * <e3> ) . 1) = ((p3 . 3) * 0 ) by A4, RVSUM_1: 44

          .= 0 ;

          

           A22: (((p3 . 3) * <e3> ) . 2) = ((p3 . 3) * 0 ) by A4, RVSUM_1: 44

          .= 0 ;

          

           A23: (((p3 . 3) * <e3> ) . 3) = ((p3 . 3) * 1) by A4, RVSUM_1: 44

          .= (p3 . 3);

          

           A24: ((((p2 . 1) * <e1> ) + ((p2 . 2) * <e2> )) + ((p2 . 3) * <e3> )) = (((p2 . 1) * <e1> ) + (((p2 . 2) * <e2> ) + ((p2 . 3) * <e3> ))) by RVSUM_1: 15

          .= (((p2 . 1) * <e1> ) + |[( 0 + 0 ), ((p2 . 2) + 0 ), ( 0 + (p2 . 3))]|) by A9, A10, A11, A12, A13, A14, Lm2

          .= ( |[(p2 . 1), 0 , 0 ]| + |[ 0 , (p2 . 2), (p2 . 3)]|) by A6, A7, A8, Th1

          .= |[((p2 . 1) + 0 ), ( 0 + (p2 . 2)), ( 0 + (p2 . 3))]| by Lm8

          .= |[(p2 . 1), (p2 . 2), (p2 . 3)]|;

          

           A25: ((((p3 . 1) * <e1> ) + ((p3 . 2) * <e2> )) + ((p3 . 3) * <e3> )) = (((p3 . 1) * <e1> ) + (((p3 . 2) * <e2> ) + ((p3 . 3) * <e3> ))) by RVSUM_1: 15

          .= (((p3 . 1) * <e1> ) + |[( 0 + 0 ), ((p3 . 2) + 0 ), ( 0 + (p3 . 3))]|) by A18, A19, A20, A21, A22, A23, Lm2

          .= ( |[(p3 . 1), 0 , 0 ]| + |[ 0 , (p3 . 2), (p3 . 3)]|) by A15, A16, A17, Th1

          .= |[((p3 . 1) + 0 ), ( 0 + (p3 . 2)), ( 0 + (p3 . 3))]| by Lm8

          .= |[(p3 . 1), (p3 . 2), (p3 . 3)]|;

          (((((((p2 . 1) * <e1> ) + ((p2 . 2) * <e2> )) + ((p2 . 3) * <e3> )) + ((p3 . 1) * <e1> )) + ((p3 . 2) * <e2> )) + ((p3 . 3) * <e3> )) = ((((((p2 . 1) * <e1> ) + (((p2 . 2) * <e2> ) + ((p2 . 3) * <e3> ))) + ((p3 . 1) * <e1> )) + ((p3 . 2) * <e2> )) + ((p3 . 3) * <e3> )) by RVSUM_1: 15

          .= ((((((p2 . 1) * <e1> ) + |[( 0 + 0 ), ((p2 . 2) + 0 ), ( 0 + (p2 . 3))]|) + ((p3 . 1) * <e1> )) + ((p3 . 2) * <e2> )) + ((p3 . 3) * <e3> )) by A9, A10, A11, A12, A13, A14, Lm2

          .= (((( |[(p2 . 1), 0 , 0 ]| + |[ 0 , (p2 . 2), (p2 . 3)]|) + ((p3 . 1) * <e1> )) + ((p3 . 2) * <e2> )) + ((p3 . 3) * <e3> )) by A6, A7, A8, Th1

          .= ((( |[((p2 . 1) + 0 ), ( 0 + (p2 . 2)), ( 0 + (p2 . 3))]| + ((p3 . 1) * <e1> )) + ((p3 . 2) * <e2> )) + ((p3 . 3) * <e3> )) by Lm8

          .= ((( |[(p2 . 1), (p2 . 2), (p2 . 3)]| + |[(p3 . 1), 0 , 0 ]|) + ((p3 . 2) * <e2> )) + ((p3 . 3) * <e3> )) by A15, A16, A17, Th1

          .= (( |[((p2 . 1) + (p3 . 1)), ((p2 . 2) + 0 ), ((p2 . 3) + 0 )]| + ((p3 . 2) * <e2> )) + ((p3 . 3) * <e3> )) by Lm8

          .= (( |[((p2 . 1) + (p3 . 1)), (p2 . 2), (p2 . 3)]| + |[ 0 , (p3 . 2), 0 ]|) + ((p3 . 3) * <e3> )) by A18, A19, A20, Th1

          .= ( |[(((p2 . 1) + (p3 . 1)) + 0 ), ((p2 . 2) + (p3 . 2)), ((p2 . 3) + 0 )]| + ((p3 . 3) * <e3> )) by Lm8

          .= ( |[((p2 . 1) + (p3 . 1)), ((p2 . 2) + (p3 . 2)), (p2 . 3)]| + |[ 0 , 0 , (p3 . 3)]|) by A21, A22, A23, Th1

          .= |[(((p2 . 1) + (p3 . 1)) + 0 ), (((p2 . 2) + (p3 . 2)) + 0 ), ((p2 . 3) + (p3 . 3))]| by Lm8;

          hence thesis by A24, A25, Lm8;

        end;

        then (((((p2 . 1) + (p3 . 1)) * <e1> ) + (((p2 . 2) + (p3 . 2)) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) = ((((((((p1 . 1) - (p3 . 1)) * <e1> ) + (((p1 . 2) - (p3 . 2)) * <e2> )) + (((p1 . 3) - (p3 . 3)) * <e3> )) + ((p3 . 1) * <e1> )) + ((p3 . 2) * <e2> )) + ((p3 . 3) * <e3> )) by A5, EUCLIDLP: 24;

        then (((((p2 . 1) + (p3 . 1)) * <e1> ) + (((p2 . 2) + (p3 . 2)) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) = (((((((p1 . 1) - (p3 . 1)) * <e1> ) + ((((p1 . 2) - (p3 . 2)) * <e2> ) + (((p1 . 3) - (p3 . 3)) * <e3> ))) + ((p3 . 1) * <e1> )) + ((p3 . 2) * <e2> )) + ((p3 . 3) * <e3> )) by RVSUM_1: 15;

        then (((((p2 . 1) + (p3 . 1)) * <e1> ) + (((p2 . 2) + (p3 . 2)) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) = (((((((p1 . 1) * <e1> ) - ((p3 . 1) * <e1> )) + ((((p1 . 2) - (p3 . 2)) * <e2> ) + (((p1 . 3) - (p3 . 3)) * <e3> ))) + ((p3 . 1) * <e1> )) + ((p3 . 2) * <e2> )) + ((p3 . 3) * <e3> )) by A1;

        then (((((p2 . 1) + (p3 . 1)) * <e1> ) + (((p2 . 2) + (p3 . 2)) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) = (((((((p1 . 1) * <e1> ) + ((((p1 . 2) - (p3 . 2)) * <e2> ) + (((p1 . 3) - (p3 . 3)) * <e3> ))) - ((p3 . 1) * <e1> )) + ((p3 . 1) * <e1> )) + ((p3 . 2) * <e2> )) + ((p3 . 3) * <e3> )) by RVSUM_1: 15;

        then (((((p2 . 1) + (p3 . 1)) * <e1> ) + (((p2 . 2) + (p3 . 2)) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) = ((((((p1 . 1) * <e1> ) + ((((p1 . 2) - (p3 . 2)) * <e2> ) + (((p1 . 3) - (p3 . 3)) * <e3> ))) + (((p3 . 1) * <e1> ) - ((p3 . 1) * <e1> ))) + ((p3 . 2) * <e2> )) + ((p3 . 3) * <e3> )) by RVSUM_1: 15;

        then (((((p2 . 1) + (p3 . 1)) * <e1> ) + (((p2 . 2) + (p3 . 2)) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) = ((((((p1 . 1) * <e1> ) + ((((p1 . 2) - (p3 . 2)) * <e2> ) + (((p1 . 3) - (p3 . 3)) * <e3> ))) + ( 0.REAL 3)) + ((p3 . 2) * <e2> )) + ((p3 . 3) * <e3> )) by RVSUM_1: 37;

        then (((((p2 . 1) + (p3 . 1)) * <e1> ) + (((p2 . 2) + (p3 . 2)) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) = (((((p1 . 1) * <e1> ) + ((((p1 . 2) - (p3 . 2)) * <e2> ) + (((p1 . 3) - (p3 . 3)) * <e3> ))) + ((p3 . 2) * <e2> )) + ((p3 . 3) * <e3> )) by EUCLID_4: 1;

        then (((((p2 . 1) + (p3 . 1)) * <e1> ) + (((p2 . 2) + (p3 . 2)) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) = ((((((p1 . 1) * <e1> ) + (((p1 . 2) - (p3 . 2)) * <e2> )) + (((p1 . 3) - (p3 . 3)) * <e3> )) + ((p3 . 2) * <e2> )) + ((p3 . 3) * <e3> )) by RVSUM_1: 15;

        then (((((p2 . 1) + (p3 . 1)) * <e1> ) + (((p2 . 2) + (p3 . 2)) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) = ((((((p1 . 1) * <e1> ) + (((p1 . 2) - (p3 . 2)) * <e2> )) + (((p1 . 3) * <e3> ) - ((p3 . 3) * <e3> ))) + ((p3 . 2) * <e2> )) + ((p3 . 3) * <e3> )) by A1;

        then (((((p2 . 1) + (p3 . 1)) * <e1> ) + (((p2 . 2) + (p3 . 2)) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) = (((((((p1 . 1) * <e1> ) + (((p1 . 2) - (p3 . 2)) * <e2> )) + ((p1 . 3) * <e3> )) - ((p3 . 3) * <e3> )) + ((p3 . 2) * <e2> )) + ((p3 . 3) * <e3> )) by RVSUM_1: 15;

        then (((((p2 . 1) + (p3 . 1)) * <e1> ) + (((p2 . 2) + (p3 . 2)) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) = (((((((p1 . 1) * <e1> ) + (((p1 . 2) - (p3 . 2)) * <e2> )) + ((p1 . 3) * <e3> )) + ((p3 . 2) * <e2> )) - ((p3 . 3) * <e3> )) + ((p3 . 3) * <e3> )) by RVSUM_1: 15;

        then (((((p2 . 1) + (p3 . 1)) * <e1> ) + (((p2 . 2) + (p3 . 2)) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) = ((((((p1 . 1) * <e1> ) + (((p1 . 2) - (p3 . 2)) * <e2> )) + ((p1 . 3) * <e3> )) + ((p3 . 2) * <e2> )) + (((p3 . 3) * <e3> ) - ((p3 . 3) * <e3> ))) by RVSUM_1: 15;

        then (((((p2 . 1) + (p3 . 1)) * <e1> ) + (((p2 . 2) + (p3 . 2)) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) = ((((((p1 . 1) * <e1> ) + (((p1 . 2) - (p3 . 2)) * <e2> )) + ((p1 . 3) * <e3> )) + ((p3 . 2) * <e2> )) + ( 0.REAL 3)) by RVSUM_1: 37;

        then (((((p2 . 1) + (p3 . 1)) * <e1> ) + (((p2 . 2) + (p3 . 2)) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) = (((((p1 . 1) * <e1> ) + (((p1 . 2) - (p3 . 2)) * <e2> )) + ((p1 . 3) * <e3> )) + ((p3 . 2) * <e2> )) by EUCLID_4: 1;

        then (((((p2 . 1) + (p3 . 1)) * <e1> ) + (((p2 . 2) + (p3 . 2)) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) = ((((p1 . 1) * <e1> ) + (((p1 . 2) - (p3 . 2)) * <e2> )) + (((p1 . 3) * <e3> ) + ((p3 . 2) * <e2> ))) by RVSUM_1: 15;

        then (((((p2 . 1) + (p3 . 1)) * <e1> ) + (((p2 . 2) + (p3 . 2)) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) = ((((p1 . 1) * <e1> ) + (((p1 . 2) * <e2> ) - ((p3 . 2) * <e2> ))) + (((p1 . 3) * <e3> ) + ((p3 . 2) * <e2> ))) by A1;

        then (((((p2 . 1) + (p3 . 1)) * <e1> ) + (((p2 . 2) + (p3 . 2)) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) = (((((p1 . 1) * <e1> ) + ((p1 . 2) * <e2> )) + ( - ((p3 . 2) * <e2> ))) + (((p1 . 3) * <e3> ) + ((p3 . 2) * <e2> ))) by RVSUM_1: 15;

        then (((((p2 . 1) + (p3 . 1)) * <e1> ) + (((p2 . 2) + (p3 . 2)) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) = ((((((p1 . 1) * <e1> ) + ((p1 . 2) * <e2> )) - ((p3 . 2) * <e2> )) + ((p1 . 3) * <e3> )) + ((p3 . 2) * <e2> )) by RVSUM_1: 15;

        then (((((p2 . 1) + (p3 . 1)) * <e1> ) + (((p2 . 2) + (p3 . 2)) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) = ((((((p1 . 1) * <e1> ) + ((p1 . 2) * <e2> )) + ((p1 . 3) * <e3> )) - ((p3 . 2) * <e2> )) + ((p3 . 2) * <e2> )) by RVSUM_1: 15;

        then (((((p2 . 1) + (p3 . 1)) * <e1> ) + (((p2 . 2) + (p3 . 2)) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) = (((((p1 . 1) * <e1> ) + ((p1 . 2) * <e2> )) + ((p1 . 3) * <e3> )) + (((p3 . 2) * <e2> ) - ((p3 . 2) * <e2> ))) by RVSUM_1: 15;

        then (((((p2 . 1) + (p3 . 1)) * <e1> ) + (((p2 . 2) + (p3 . 2)) * <e2> )) + (((p2 . 3) + (p3 . 3)) * <e3> )) = (((((p1 . 1) * <e1> ) + ((p1 . 2) * <e2> )) + ((p1 . 3) * <e3> )) + ( 0.REAL 3)) by RVSUM_1: 37;

        hence thesis by EUCLID_4: 1;

      end;

      hence thesis;

    end;

    definition

      let f1,f2,f3 be PartFunc of REAL , REAL ;

      :: EUCLID_8:def5

      func VFunc (f1,f2,f3) -> Function of REAL , ( REAL 3) means

      : Def5: for t holds (it . t) = |[(f1 . t), (f2 . t), (f3 . t)]|;

      existence

      proof

        defpred P[ object, object] means $2 = |[(f1 . $1), (f2 . $1), (f3 . $1)]|;

        

         A1: for x be Element of REAL holds ex y be Element of ( REAL 3) st P[x, y];

        consider F be Function of REAL , ( REAL 3) such that

         A2: for t be Element of REAL holds P[t, (F . t)] from FUNCT_2:sch 3( A1);

        

         A3: for t be Real holds P[t, (F . t)]

        proof

          let t be Real;

          reconsider t as Element of REAL by XREAL_0:def 1;

           P[t, (F . t)] by A2;

          hence thesis;

        end;

        take F;

        thus thesis by A3;

      end;

      uniqueness

      proof

        let F,G be Function of REAL , ( REAL 3);

        assume that

         A4: for t holds (F . t) = |[(f1 . t), (f2 . t), (f3 . t)]| and

         A5: for t holds (G . t) = |[(f1 . t), (f2 . t), (f3 . t)]|;

        now

          let t be Element of REAL ;

          (F . t) = |[(f1 . t), (f2 . t), (f3 . t)]| by A4;

          hence (F . t) = (G . t) by A5;

        end;

        hence thesis by FUNCT_2: 63;

      end;

    end

    theorem :: EUCLID_8:48

    (( VFunc (f1,f2,f3)) . t) = ((((f1 . t) * <e1> ) + ((f2 . t) * <e2> )) + ((f3 . t) * <e3> ))

    proof

      

       A1: ( <e1> . 1) = 1 & ( <e1> . 2) = 0 & ( <e1> . 3) = 0 by FINSEQ_1: 45;

      

       A2: ( <e2> . 1) = 0 & ( <e2> . 2) = 1 & ( <e2> . 3) = 0 by FINSEQ_1: 45;

      

       A3: ( <e3> . 1) = 0 & ( <e3> . 2) = 0 & ( <e3> . 3) = 1 by FINSEQ_1: 45;

      

       A4: ((((f1 . t) * <e1> ) + ((f2 . t) * <e2> )) + ((f3 . t) * <e3> )) = (( |[((f1 . t) * 1), ((f1 . t) * 0 ), ((f1 . t) * 0 )]| + ((f2 . t) * <e2> )) + ((f3 . t) * <e3> )) by A1, Lm1

      .= (( |[(f1 . t), 0 , 0 ]| + |[((f2 . t) * 0 ), ((f2 . t) * 1), ((f2 . t) * 0 )]|) + ((f3 . t) * <e3> )) by A2, Lm1

      .= (( |[(f1 . t), 0 , 0 ]| + |[ 0 , (f2 . t), 0 ]|) + |[((f3 . t) * 0 ), ((f3 . t) * 0 ), ((f3 . t) * 1)]|) by A3, Lm1

      .= (( |[(f1 . t), 0 , 0 ]| + |[ 0 , (f2 . t), 0 ]|) + |[ 0 , 0 , (f3 . t)]|);

      

       A5: ( |[(f1 . t), 0 , 0 ]| . 1) = (f1 . t) & ( |[(f1 . t), 0 , 0 ]| . 2) = 0 & ( |[(f1 . t), 0 , 0 ]| . 3) = 0 by FINSEQ_1: 45;

      

       A6: ( |[ 0 , (f2 . t), 0 ]| . 1) = 0 & ( |[ 0 , (f2 . t), 0 ]| . 2) = (f2 . t) & ( |[ 0 , (f2 . t), 0 ]| . 3) = 0 by FINSEQ_1: 45;

      

       A7: ( |[ 0 , 0 , (f3 . t)]| . 1) = 0 & ( |[ 0 , 0 , (f3 . t)]| . 2) = 0 & ( |[ 0 , 0 , (f3 . t)]| . 3) = (f3 . t) by FINSEQ_1: 45;

      

       A8: ((((f1 . t) * <e1> ) + ((f2 . t) * <e2> )) + ((f3 . t) * <e3> )) = ( |[((f1 . t) + 0 ), ( 0 + (f2 . t)), ( 0 + 0 )]| + |[ 0 , 0 , (f3 . t)]|) by A4, A5, A6, Lm2

      .= ( |[(f1 . t), (f2 . t), 0 ]| + |[ 0 , 0 , (f3 . t)]|);

      ( |[(f1 . t), (f2 . t), 0 ]| . 1) = (f1 . t) & ( |[(f1 . t), (f2 . t), 0 ]| . 2) = (f2 . t) & ( |[(f1 . t), (f2 . t), 0 ]| . 3) = 0 by FINSEQ_1: 45;

      

      then ((((f1 . t) * <e1> ) + ((f2 . t) * <e2> )) + ((f3 . t) * <e3> )) = |[((f1 . t) + 0 ), ((f2 . t) + 0 ), ( 0 + (f3 . t))]| by A7, A8, Lm2

      .= |[(f1 . t), (f2 . t), (f3 . t)]|;

      hence (( VFunc (f1,f2,f3)) . t) = ((((f1 . t) * <e1> ) + ((f2 . t) * <e2> )) + ((f3 . t) * <e3> )) by Def5;

    end;

    theorem :: EUCLID_8:49

    

     Th41: p = (( VFunc (f1,f2,f3)) . t) iff (p . 1) = (f1 . t) & (p . 2) = (f2 . t) & (p . 3) = (f3 . t)

    proof

      thus p = (( VFunc (f1,f2,f3)) . t) implies (p . 1) = (f1 . t) & (p . 2) = (f2 . t) & (p . 3) = (f3 . t)

      proof

        assume p = (( VFunc (f1,f2,f3)) . t);

        then p = |[(f1 . t), (f2 . t), (f3 . t)]| by Def5;

        hence thesis by FINSEQ_1: 45;

      end;

      assume (p . 1) = (f1 . t) & (p . 2) = (f2 . t) & (p . 3) = (f3 . t);

      then p = |[(f1 . t), (f2 . t), (f3 . t)]| by Th1;

      hence p = (( VFunc (f1,f2,f3)) . t) by Def5;

    end;

    theorem :: EUCLID_8:50

    

     Th42: ( len p) = 3 & ( dom p) = ( Seg 3)

    proof

      p = |[(p . 1), (p . 2), (p . 3)]| by Th1;

      hence thesis by FINSEQ_1: 45, FINSEQ_1: 89;

    end;

    theorem :: EUCLID_8:51

    

     Th43: ( mlt (p,q)) = <*((p . 1) * (q . 1)), ((p . 2) * (q . 2)), ((p . 3) * (q . 3))*>

    proof

      ( len p) = 3 & ( len q) = 3 by Th42;

      hence thesis by EUCLID_5: 28;

    end;

    theorem :: EUCLID_8:52

    (r * p) = |[(r * (p . 1)), (r * (p . 2)), (r * (p . 3))]| by Lm1;

    ::$Canceled

    theorem :: EUCLID_8:54

    ( len ( - p)) = ( len p)

    proof

      

       A1: ( len p) = 3 by Th42;

      ( - p) = |[(( - p) . 1), (( - p) . 2), (( - p) . 3)]| by Th1;

      hence thesis by A1, FINSEQ_1: 45;

    end;

    theorem :: EUCLID_8:55

    (p + q) = |[((p . 1) + (q . 1)), ((p . 2) + (q . 2)), ((p . 3) + (q . 3))]| by Lm2;

    theorem :: EUCLID_8:56

    (p = (( VFunc (f1,f2,f3)) . t1) & q = (( VFunc (g1,g2,g3)) . t2) & p = q) implies ((f1 . t1) = (g1 . t2) & (f2 . t1) = (g2 . t2) & (f3 . t1) = (g3 . t2))

    proof

      assume

       A1: p = (( VFunc (f1,f2,f3)) . t1) & q = (( VFunc (g1,g2,g3)) . t2) & p = q;

      then

       A2: (p . 1) = (f1 . t1) & (q . 1) = (g1 . t2) by Th41;

      

       A3: (p . 2) = (f2 . t1) & (q . 2) = (g2 . t2) by A1, Th41;

      (p . 3) = (f3 . t1) & (q . 3) = (g3 . t2) by A1, Th41;

      hence thesis by A1, A2, A3;

    end;

    theorem :: EUCLID_8:57

    ((f1 . t1) = (g1 . t2) & (f2 . t1) = (g2 . t2) & (f3 . t1) = (g3 . t2)) implies (( VFunc (f1,f2,f3)) . t1) = (( VFunc (g1,g2,g3)) . t2)

    proof

      assume

       A1: (f1 . t1) = (g1 . t2) & (f2 . t1) = (g2 . t2) & (f3 . t1) = (g3 . t2);

      set p = |[(f1 . t1), (f2 . t1), (f3 . t1)]|;

      set q = |[(g1 . t2), (g2 . t2), (g3 . t2)]|;

      p = (( VFunc (f1,f2,f3)) . t1) & q = (( VFunc (g1,g2,g3)) . t2) by Def5;

      hence thesis by A1;

    end;

    theorem :: EUCLID_8:58

    

     Th49: for r be Real holds (r * p) = |[(r * (p . 1)), (r * (p . 2)), (r * (p . 3))]|

    proof

      let r be Real;

      

       A1: ((r * p) . 1) = (r * (p . 1)) by RVSUM_1: 44;

      

       A2: ((r * p) . 2) = (r * (p . 2)) by RVSUM_1: 44;

      ((r * p) . 3) = (r * (p . 3)) by RVSUM_1: 44;

      hence (r * p) = |[(r * (p . 1)), (r * (p . 2)), (r * (p . 3))]| by A1, A2, Th1;

    end;

    theorem :: EUCLID_8:59

    

     Th50: for r be Real holds (r * |[x, y, z]|) = |[(r * x), (r * y), (r * z)]|

    proof

      let r be Real;

      set p = |[x, y, z]|;

      (r * p) = |[(r * (p . 1)), (r * (p . 2)), (r * (p . 3))]| by Th49

      .= |[(r * x), (r * (p . 2)), (r * (p . 3))]| by FINSEQ_1: 45

      .= |[(r * x), (r * y), (r * (p . 3))]| by FINSEQ_1: 45

      .= |[(r * x), (r * y), (r * z)]| by FINSEQ_1: 45;

      hence thesis;

    end;

    theorem :: EUCLID_8:60

    

     Th51: ( - p) = |[( - (p . 1)), ( - (p . 2)), ( - (p . 3))]|

    proof

      reconsider r = ( - 1) as Element of REAL by XREAL_0:def 1;

      (r * p) = |[(( - 1) * (p . 1)), (( - 1) * (p . 2)), (( - 1) * (p . 3))]| by Th49

      .= |[( - (p . 1)), ( - (p . 2)), ( - (p . 3))]|;

      hence thesis;

    end;

    theorem :: EUCLID_8:61

    

     Th52: (( - p) . 1) = ( - (p . 1)) & (( - p) . 2) = ( - (p . 2)) & (( - p) . 3) = ( - (p . 3))

    proof

      ( - p) = |[( - (p . 1)), ( - (p . 2)), ( - (p . 3))]| by Th51;

      hence thesis by FINSEQ_1: 45;

    end;

    theorem :: EUCLID_8:62

    (p1 - p2) = |[((p1 . 1) - (p2 . 1)), ((p1 . 2) - (p2 . 2)), ((p1 . 3) - (p2 . 3))]|

    proof

      

       A1: (( - p2) . 1) = ( - (p2 . 1)) by Th52;

      

       A2: (( - p2) . 2) = ( - (p2 . 2)) by Th52;

      (( - p2) . 3) = ( - (p2 . 3)) by Th52;

      then (p1 + ( - p2)) = |[((p1 . 1) + ( - (p2 . 1))), ((p1 . 2) + ( - (p2 . 2))), ((p1 . 3) + ( - (p2 . 3)))]| by A1, A2, Lm2;

      hence thesis;

    end;

    theorem :: EUCLID_8:63

     |(p, q)| = ((((p . 1) * (q . 1)) + ((p . 2) * (q . 2))) + ((p . 3) * (q . 3))) by Lm5;

    theorem :: EUCLID_8:64

    

     Th55: |(p, p)| = ((((p . 1) ^2 ) + ((p . 2) ^2 )) + ((p . 3) ^2 ))

    proof

      p = |[(p . 1), (p . 2), (p . 3)]| by Th1;

      hence thesis by Lm12;

    end;

    theorem :: EUCLID_8:65

    

     Th56: |.p.| = ( sqrt ((((p . 1) ^2 ) + ((p . 2) ^2 )) + ((p . 3) ^2 )))

    proof

       |.p.| = ( sqrt ( Sum <*((p . 1) * (p . 1)), ((p . 2) * (p . 2)), ((p . 3) * (p . 3))*>)) by Th43

      .= ( sqrt ((((p . 1) ^2 ) + ((p . 2) ^2 )) + ((p . 3) ^2 ))) by RVSUM_1: 78;

      hence thesis;

    end;

    theorem :: EUCLID_8:66

     |.(r * p).| = ( |.r.| * ( sqrt ((((p . 1) ^2 ) + ((p . 2) ^2 )) + ((p . 3) ^2 ))))

    proof

       |.(r * p).| = ( |.r.| * |.p.|) by EUCLID: 11

      .= ( |.r.| * ( sqrt ((((p . 1) ^2 ) + ((p . 2) ^2 )) + ((p . 3) ^2 )))) by Th56;

      hence thesis;

    end;

    theorem :: EUCLID_8:67

    ((r1 * p) + (r2 * p)) = ((r1 + r2) * |[(p . 1), (p . 2), (p . 3)]|)

    proof

      ((r1 * p) + (r2 * p)) = ((r1 + r2) * p) by RVSUM_1: 50;

      hence thesis by Th1;

    end;

    theorem :: EUCLID_8:68

     |((r * p1), p2)| = (r * |(p1, p2)|) by RVSUM_1: 131;

    theorem :: EUCLID_8:69

    ((r1 * p) - (r2 * p)) = ((r1 - r2) * |[(p . 1), (p . 2), (p . 3)]|)

    proof

      ((r1 * R) - (r2 * R)) = ((r1 - r2) * R)

      proof

        ((r1 * R) - (r2 * R)) = ((r1 * R) + ((( - 1) * r2) * R)) by RVSUM_1: 49

        .= ((r1 + ( - r2)) * R) by RVSUM_1: 50;

        hence thesis;

      end;

      then ((r1 * p) - (r2 * p)) = ((r1 - r2) * p);

      hence thesis by Th1;

    end;

    theorem :: EUCLID_8:70

     |((r * p), q)| = (r * ((((p . 1) * (q . 1)) + ((p . 2) * (q . 2))) + ((p . 3) * (q . 3))))

    proof

       |((r * p), q)| = (r * |(p, q)|) by RVSUM_1: 131;

      hence thesis by Lm5;

    end;

    theorem :: EUCLID_8:71

     |(p, ( 0.REAL 3))| = 0

    proof

      ( 0.REAL 3) = |[ 0 , 0 , 0 ]| by FINSEQ_2: 62;

      hence thesis by Th13;

    end;

    theorem :: EUCLID_8:72

     |(( - p1), p2)| = ( - |(p1, p2)|) by RVSUM_1: 132;

    theorem :: EUCLID_8:73

     |(( - p1), ( - p2))| = |(p1, p2)| by RVSUM_1: 133;

    theorem :: EUCLID_8:74

     |((p1 - p2), q)| = ( |(p1, q)| - |(p2, q)|) by RVSUM_1: 134;

    theorem :: EUCLID_8:75

     |((p1 + p2), q)| = ( |(p1, q)| + |(p2, q)|) by RVSUM_1: 130;

    theorem :: EUCLID_8:76

     |(((r1 * p1) + (r2 * p2)), q)| = ((r1 * |(p1, q)|) + (r2 * |(p2, q)|)) by RVSUM_1: 135;

    theorem :: EUCLID_8:77

     |((p1 + p2), (q1 + q2))| = ((( |(p1, q1)| + |(p1, q2)|) + |(p2, q1)|) + |(p2, q2)|) by RVSUM_1: 136;

    theorem :: EUCLID_8:78

     |((p1 - p2), (q1 - q2))| = ((( |(p1, q1)| - |(p1, q2)|) - |(p2, q1)|) + |(p2, q2)|) by RVSUM_1: 137;

    theorem :: EUCLID_8:79

    

     Th70: |(p, p)| = 0 iff p = ( 0.REAL 3)

    proof

      thus |(p, p)| = 0 implies p = ( 0.REAL 3)

      proof

        assume |(p, p)| = 0 ;

        then ( Sum ( sqr p)) = 0 ;

        hence thesis by RVSUM_1: 91;

      end;

      assume p = ( 0.REAL 3);

      then

       A1: p = |[ 0 , 0 , 0 ]| by FINSEQ_2: 62;

      then

       A2: (p . 1) = 0 by FINSEQ_1: 45;

      

       A3: (p . 2) = 0 by A1, FINSEQ_1: 45;

      (p . 3) = 0 by A1, FINSEQ_1: 45;

      then |(p, p)| = ((( 0 ^2 ) + ( 0 ^2 )) + ( 0 ^2 )) by A2, A3, Th55;

      hence |(p, p)| = 0 ;

    end;

    theorem :: EUCLID_8:80

     |.p.| = 0 iff p = ( 0.REAL 3)

    proof

      thus |.p.| = 0 implies p = ( 0.REAL 3)

      proof

        assume

         A1: |.p.| = 0 ;

         |(p, p)| >= 0 by RVSUM_1: 119;

        then (( sqrt |(p, p)|) ^2 ) = |(p, p)| by SQUARE_1:def 2;

        hence thesis by A1, Th70;

      end;

      assume p = ( 0.REAL 3);

      then

       A2: p = |[ 0 , 0 , 0 ]| by FINSEQ_2: 62;

      then

       A3: (p . 1) = 0 by FINSEQ_1: 45;

      

       A4: (p . 2) = 0 by A2, FINSEQ_1: 45;

      (p . 3) = 0 by A2, FINSEQ_1: 45;

      then |(p, p)| = ((( 0 ^2 ) + ( 0 ^2 )) + ( 0 ^2 )) by A3, A4, Th55;

      hence thesis by SQUARE_1: 17;

    end;

    theorem :: EUCLID_8:81

     |((p - q), (p - q))| = (( |(p, p)| - (2 * |(p, q)|)) + |(q, q)|) by RVSUM_1: 139;

    theorem :: EUCLID_8:82

     |((p + q), (p + q))| = (( |(p, p)| + (2 * |(p, q)|)) + |(q, q)|) by RVSUM_1: 138;

    theorem :: EUCLID_8:83

    

     Th74: (p1 <X> p2) = ( - (p2 <X> p1))

    proof

      ( - (p2 <X> p1)) = |[(( - 1) * (((p2 . 2) * (p1 . 3)) - ((p2 . 3) * (p1 . 2)))), (( - 1) * (((p2 . 3) * (p1 . 1)) - ((p2 . 1) * (p1 . 3)))), (( - 1) * (((p2 . 1) * (p1 . 2)) - ((p2 . 2) * (p1 . 1))))]| by Th50

      .= (p1 <X> p2);

      hence thesis;

    end;

    theorem :: EUCLID_8:84

    

     Th75: ( |[x1, x2, x3]| <X> |[y1, y2, y3]|) = |[((x2 * y3) - (x3 * y2)), ((x3 * y1) - (x1 * y3)), ((x1 * y2) - (x2 * y1))]|

    proof

      set p1 = |[x1, x2, x3]|;

      

       A1: (p1 . 1) = x1 & (p1 . 2) = x2 & (p1 . 3) = x3 by FINSEQ_1: 45;

      set p2 = |[y1, y2, y3]|;

      (p2 . 1) = y1 & (p2 . 2) = y2 & (p2 . 3) = y3 by FINSEQ_1: 45;

      hence thesis by A1;

    end;

    theorem :: EUCLID_8:85

    ((r * p1) <X> p2) = (r * (p1 <X> p2)) & ((r * p1) <X> p2) = (p1 <X> (r * p2))

    proof

      

       A1: ((r * p1) <X> p2) = ( |[(r * (p1 . 1)), (r * (p1 . 2)), (r * (p1 . 3))]| <X> p2) by Th49

      .= ( |[(r * (p1 . 1)), (r * (p1 . 2)), (r * (p1 . 3))]| <X> |[(p2 . 1), (p2 . 2), (p2 . 3)]|) by Th1

      .= |[(((r * (p1 . 2)) * (p2 . 3)) - ((r * (p1 . 3)) * (p2 . 2))), (((r * (p1 . 3)) * (p2 . 1)) - ((r * (p1 . 1)) * (p2 . 3))), (((r * (p1 . 1)) * (p2 . 2)) - ((r * (p1 . 2)) * (p2 . 1)))]| by Th75;

      

      then

       A2: ((r * p1) <X> p2) = |[(r * (((p1 . 2) * (p2 . 3)) - ((p1 . 3) * (p2 . 2)))), (r * (((p1 . 3) * (p2 . 1)) - ((p1 . 1) * (p2 . 3)))), (r * (((p1 . 1) * (p2 . 2)) - ((p1 . 2) * (p2 . 1))))]|

      .= (r * (p1 <X> p2)) by Th50;

      ((r * p1) <X> p2) = |[(((p1 . 2) * (r * (p2 . 3))) - ((p1 . 3) * (r * (p2 . 2)))), (((p1 . 3) * (r * (p2 . 1))) - ((p1 . 1) * (r * (p2 . 3)))), (((p1 . 1) * (r * (p2 . 2))) - ((p1 . 2) * (r * (p2 . 1))))]| by A1

      .= ( |[(p1 . 1), (p1 . 2), (p1 . 3)]| <X> |[(r * (p2 . 1)), (r * (p2 . 2)), (r * (p2 . 3))]|) by Th75

      .= (p1 <X> |[(r * (p2 . 1)), (r * (p2 . 2)), (r * (p2 . 3))]|) by Th1

      .= (p1 <X> (r * p2)) by Th49;

      hence thesis by A2;

    end;

    theorem :: EUCLID_8:86

    

     Th77: (p1 <X> (p2 + p3)) = ((p1 <X> p2) + (p1 <X> p3))

    proof

      

       A1: (p2 + p3) = |[((p2 . 1) + (p3 . 1)), ((p2 . 2) + (p3 . 2)), ((p2 . 3) + (p3 . 3))]| by Lm2;

      then

       A2: ((p2 + p3) . 1) = ((p2 . 1) + (p3 . 1)) by FINSEQ_1: 45;

      

       A3: ((p2 + p3) . 2) = ((p2 . 2) + (p3 . 2)) by A1, FINSEQ_1: 45;

      

       A4: ((p2 + p3) . 3) = ((p2 . 3) + (p3 . 3)) by A1, FINSEQ_1: 45;

      ((p1 <X> p2) + (p1 <X> p3)) = |[((((p1 . 2) * (p2 . 3)) - ((p1 . 3) * (p2 . 2))) + (((p1 . 2) * (p3 . 3)) - ((p1 . 3) * (p3 . 2)))), ((((p1 . 3) * (p2 . 1)) - ((p1 . 1) * (p2 . 3))) + (((p1 . 3) * (p3 . 1)) - ((p1 . 1) * (p3 . 3)))), ((((p1 . 1) * (p2 . 2)) - ((p1 . 2) * (p2 . 1))) + (((p1 . 1) * (p3 . 2)) - ((p1 . 2) * (p3 . 1))))]| by Lm8

      .= |[(((((p1 . 2) * (p2 . 3)) - ((p1 . 3) * (p2 . 2))) + ((p1 . 2) * (p3 . 3))) - ((p1 . 3) * (p3 . 2))), (((((p1 . 3) * (p2 . 1)) - ((p1 . 1) * (p2 . 3))) + ((p1 . 3) * (p3 . 1))) - ((p1 . 1) * (p3 . 3))), (((((p1 . 1) * (p2 . 2)) - ((p1 . 2) * (p2 . 1))) + ((p1 . 1) * (p3 . 2))) - ((p1 . 2) * (p3 . 1)))]|;

      hence thesis by A2, A3, A4;

    end;

    theorem :: EUCLID_8:87

    

     Th78: ((p1 + p2) <X> p3) = ((p1 <X> p3) + (p2 <X> p3))

    proof

      ((p1 + p2) <X> p3) = ( - (p3 <X> (p1 + p2))) by Th74

      .= ( - ((p3 <X> p1) + (p3 <X> p2))) by Th77

      .= ( - ((p3 <X> p1) - (p2 <X> p3))) by Th74

      .= (( - (p3 <X> p1)) + (p2 <X> p3)) by RVSUM_1: 36;

      hence thesis by Th74;

    end;

    theorem :: EUCLID_8:88

    ((p1 + p2) <X> (q1 + q2)) = ((((p1 <X> q1) + (p1 <X> q2)) + (p2 <X> q1)) + (p2 <X> q2))

    proof

      ((p1 + p2) <X> (q1 + q2)) = ((p1 <X> (q1 + q2)) + (p2 <X> (q1 + q2))) by Th78;

      then ((p1 + p2) <X> (q1 + q2)) = (((p1 <X> q1) + (p1 <X> q2)) + (p2 <X> (q1 + q2))) by Th77;

      then ((p1 + p2) <X> (q1 + q2)) = (((p1 <X> q1) + (p1 <X> q2)) + ((p2 <X> q1) + (p2 <X> q2))) by Th77;

      hence thesis by RVSUM_1: 15;

    end;

    theorem :: EUCLID_8:89

    (p1 <X> (p2 <X> p3)) = (( |(p1, p3)| * p2) - ( |(p1, p2)| * p3))

    proof

      

       A1: p2 = |[(p2 . 1), (p2 . 2), (p2 . 3)]| by Th1;

      

       A2: p3 = |[(p3 . 1), (p3 . 2), (p3 . 3)]| by Th1;

      (p1 <X> (p2 <X> p3)) = ( |[(p1 . 1), (p1 . 2), (p1 . 3)]| <X> |[(((p2 . 2) * (p3 . 3)) - ((p2 . 3) * (p3 . 2))), (((p2 . 3) * (p3 . 1)) - ((p2 . 1) * (p3 . 3))), (((p2 . 1) * (p3 . 2)) - ((p2 . 2) * (p3 . 1)))]|) by Th1;

      then (p1 <X> (p2 <X> p3)) = |[(((p1 . 2) * (((p2 . 1) * (p3 . 2)) - ((p2 . 2) * (p3 . 1)))) - ((p1 . 3) * (((p2 . 3) * (p3 . 1)) - ((p2 . 1) * (p3 . 3))))), (((p1 . 3) * (((p2 . 2) * (p3 . 3)) - ((p2 . 3) * (p3 . 2)))) - ((p1 . 1) * (((p2 . 1) * (p3 . 2)) - ((p2 . 2) * (p3 . 1))))), (((p1 . 1) * (((p2 . 3) * (p3 . 1)) - ((p2 . 1) * (p3 . 3)))) - ((p1 . 2) * (((p2 . 2) * (p3 . 3)) - ((p2 . 3) * (p3 . 2)))))]| by Th75;

      then

       A3: (p1 <X> (p2 <X> p3)) = |[((((((p1 . 2) * (p3 . 2)) + ((p1 . 3) * (p3 . 3))) + ((p1 . 1) * (p3 . 1))) * (p2 . 1)) - (((((p1 . 2) * (p2 . 2)) + ((p1 . 3) * (p2 . 3))) + ((p1 . 1) * (p2 . 1))) * (p3 . 1))), ((((((p1 . 3) * (p3 . 3)) + ((p1 . 1) * (p3 . 1))) + ((p1 . 2) * (p3 . 2))) * (p2 . 2)) - (((((p1 . 3) * (p2 . 3)) + ((p1 . 1) * (p2 . 1))) + ((p1 . 2) * (p2 . 2))) * (p3 . 2))), ((((((p1 . 1) * (p3 . 1)) + ((p1 . 2) * (p3 . 2))) + ((p1 . 3) * (p3 . 3))) * (p2 . 3)) - (((((p1 . 1) * (p2 . 1)) + ((p1 . 2) * (p2 . 2))) + ((p1 . 3) * (p2 . 3))) * (p3 . 3)))]|;

       |(p1, p3)| = ((((p1 . 1) * (p3 . 1)) + ((p1 . 2) * (p3 . 2))) + ((p1 . 3) * (p3 . 3))) by Lm5;

      then

       A4: (p1 <X> (p2 <X> p3)) = ( |[( |(p1, p3)| * (p2 . 1)), ( |(p1, p3)| * (p2 . 2)), ( |(p1, p3)| * (p2 . 3))]| - |[(((((p1 . 1) * (p2 . 1)) + ((p1 . 2) * (p2 . 2))) + ((p1 . 3) * (p2 . 3))) * (p3 . 1)), (((((p1 . 1) * (p2 . 1)) + ((p1 . 2) * (p2 . 2))) + ((p1 . 3) * (p2 . 3))) * (p3 . 2)), (((((p1 . 1) * (p2 . 1)) + ((p1 . 2) * (p2 . 2))) + ((p1 . 3) * (p2 . 3))) * (p3 . 3))]|) by A3, Lm11;

       |(p1, p2)| = ((((p1 . 1) * (p2 . 1)) + ((p1 . 2) * (p2 . 2))) + ((p1 . 3) * (p2 . 3))) by Lm5;

      then (p1 <X> (p2 <X> p3)) = (( |(p1, p3)| * |[(p2 . 1), (p2 . 2), (p2 . 3)]|) - |[( |(p1, p2)| * (p3 . 1)), ( |(p1, p2)| * (p3 . 2)), ( |(p1, p2)| * (p3 . 3))]|) by A4, Th50;

      hence thesis by A1, A2, Th50;

    end;

    definition

      let p1, p2, p3;

      :: EUCLID_8:def6

      func |{p1,p2,p3}| -> Real equals |(p1, (p2 <X> p3))|;

      coherence ;

    end

    theorem :: EUCLID_8:90

     |{p1, p1, p2}| = 0

    proof

      

       A1: ((p1 <X> p2) . 1) = (((p1 . 2) * (p2 . 3)) - ((p1 . 3) * (p2 . 2))) by FINSEQ_1: 45;

      

       A2: ((p1 <X> p2) . 2) = (((p1 . 3) * (p2 . 1)) - ((p1 . 1) * (p2 . 3))) by FINSEQ_1: 45;

      ((p1 <X> p2) . 3) = (((p1 . 1) * (p2 . 2)) - ((p1 . 2) * (p2 . 1))) by FINSEQ_1: 45;

      

      then |{p1, p1, p2}| = ((((p1 . 1) * (((p1 . 2) * (p2 . 3)) - ((p1 . 3) * (p2 . 2)))) + ((p1 . 2) * (((p1 . 3) * (p2 . 1)) - ((p1 . 1) * (p2 . 3))))) + ((p1 . 3) * (((p1 . 1) * (p2 . 2)) - ((p1 . 2) * (p2 . 1))))) by A2, A1, Lm5

      .= 0 ;

      hence thesis;

    end;

    theorem :: EUCLID_8:91

     |{p2, p1, p2}| = 0

    proof

      

       A1: ((p1 <X> p2) . 1) = (((p1 . 2) * (p2 . 3)) - ((p1 . 3) * (p2 . 2))) by FINSEQ_1: 45;

      

       A2: ((p1 <X> p2) . 2) = (((p1 . 3) * (p2 . 1)) - ((p1 . 1) * (p2 . 3))) by FINSEQ_1: 45;

      ((p1 <X> p2) . 3) = (((p1 . 1) * (p2 . 2)) - ((p1 . 2) * (p2 . 1))) by FINSEQ_1: 45;

      

      then |{p2, p1, p2}| = ((((p2 . 1) * (((p1 . 2) * (p2 . 3)) - ((p1 . 3) * (p2 . 2)))) + ((p2 . 2) * (((p1 . 3) * (p2 . 1)) - ((p1 . 1) * (p2 . 3))))) + ((p2 . 3) * (((p1 . 1) * (p2 . 2)) - ((p1 . 2) * (p2 . 1))))) by A2, A1, Lm5

      .= 0 ;

      hence thesis;

    end;

    theorem :: EUCLID_8:92

     |{p1, p2, p2}| = 0

    proof

       |{p1, p2, p2}| = ((((p1 . 1) * ((p2 <X> p2) . 1)) + ((p1 . 2) * ((p2 <X> p2) . 2))) + ((p1 . 3) * ((p2 <X> p2) . 3))) by Lm5

      .= ((((p1 . 1) * (((p2 . 2) * (p2 . 3)) - ((p2 . 3) * (p2 . 2)))) + ((p1 . 2) * ((p2 <X> p2) . 2))) + ((p1 . 3) * ((p2 <X> p2) . 3))) by FINSEQ_1: 45

      .= ((((p1 . 1) * (((p2 . 2) * (p2 . 3)) - ((p2 . 3) * (p2 . 2)))) + ((p1 . 2) * (((p2 . 3) * (p2 . 1)) - ((p2 . 1) * (p2 . 3))))) + ((p1 . 3) * ((p2 <X> p2) . 3))) by FINSEQ_1: 45

      .= (((((p1 . 1) * ((p2 . 2) * (p2 . 3))) - ((p1 . 1) * ((p2 . 3) * (p2 . 2)))) + ((p1 . 2) * (((p2 . 3) * (p2 . 1)) - ((p2 . 1) * (p2 . 3))))) + ((p1 . 3) * (((p2 . 1) * (p2 . 2)) - ((p2 . 2) * (p2 . 1))))) by FINSEQ_1: 45

      .= (( 0 - ((p2 . 2) * ((p2 . 1) * (p2 . 3)))) + ((p2 . 2) * ((p2 . 1) * (p2 . 3))));

      hence thesis;

    end;

    theorem :: EUCLID_8:93

    

     Th84: |{p1, p2, p3}| = |{p2, p3, p1}|

    proof

       |{p1, p2, p3}| = |( |[(p1 . 1), (p1 . 2), (p1 . 3)]|, |[(((p2 . 2) * (p3 . 3)) - ((p2 . 3) * (p3 . 2))), (((p2 . 3) * (p3 . 1)) - ((p2 . 1) * (p3 . 3))), (((p2 . 1) * (p3 . 2)) - ((p2 . 2) * (p3 . 1)))]|)| by Th1

      .= ((((p1 . 1) * (((p2 . 2) * (p3 . 3)) - ((p2 . 3) * (p3 . 2)))) + ((p1 . 2) * (((p2 . 3) * (p3 . 1)) - ((p2 . 1) * (p3 . 3))))) + ((p1 . 3) * (((p2 . 1) * (p3 . 2)) - ((p2 . 2) * (p3 . 1))))) by EUCLID_5: 30

      .= ((((p2 . 1) * (((p3 . 2) * (p1 . 3)) - ((p3 . 3) * (p1 . 2)))) + ((p2 . 2) * (((p3 . 3) * (p1 . 1)) - ((p3 . 1) * (p1 . 3))))) + ((p2 . 3) * (((p3 . 1) * (p1 . 2)) - ((p3 . 2) * (p1 . 1)))))

      .= |( |[(p2 . 1), (p2 . 2), (p2 . 3)]|, |[(((p3 . 2) * (p1 . 3)) - ((p3 . 3) * (p1 . 2))), (((p3 . 3) * (p1 . 1)) - ((p3 . 1) * (p1 . 3))), (((p3 . 1) * (p1 . 2)) - ((p3 . 2) * (p1 . 1)))]|)| by EUCLID_5: 30

      .= |(p2, (p3 <X> p1))| by Th1;

      hence thesis;

    end;

    theorem :: EUCLID_8:94

     |{p1, p2, p3}| = |((p1 <X> p2), p3)|

    proof

       |{p1, p2, p3}| = |( |[(p1 . 1), (p1 . 2), (p1 . 3)]|, |[(((p2 . 2) * (p3 . 3)) - ((p2 . 3) * (p3 . 2))), (((p2 . 3) * (p3 . 1)) - ((p2 . 1) * (p3 . 3))), (((p2 . 1) * (p3 . 2)) - ((p2 . 2) * (p3 . 1)))]|)| by Th1

      .= ((((p1 . 1) * (((p2 . 2) * (p3 . 3)) - ((p2 . 3) * (p3 . 2)))) + ((p1 . 2) * (((p2 . 3) * (p3 . 1)) - ((p2 . 1) * (p3 . 3))))) + ((p1 . 3) * (((p2 . 1) * (p3 . 2)) - ((p2 . 2) * (p3 . 1))))) by EUCLID_5: 30

      .= (((((p2 . 2) * ((p1 . 1) * (p3 . 3))) - ((p2 . 3) * ((p1 . 1) * (p3 . 2)))) + (((p2 . 3) * ((p1 . 2) * (p3 . 1))) - ((p2 . 1) * ((p1 . 2) * (p3 . 3))))) + (((p2 . 1) * ((p1 . 3) * (p3 . 2))) - ((p2 . 2) * ((p1 . 3) * (p3 . 1)))));

      

      then |{p1, p2, p3}| = ((((((p2 . 3) * (p1 . 2)) - ((p2 . 2) * (p1 . 3))) * (p3 . 1)) + ((((p2 . 1) * (p1 . 3)) - ((p2 . 3) * (p1 . 1))) * (p3 . 2))) + ((((p2 . 2) * (p1 . 1)) - ((p2 . 1) * (p1 . 2))) * (p3 . 3)))

      .= |((p1 <X> p2), |[(p3 . 1), (p3 . 2), (p3 . 3)]|)| by EUCLID_5: 30

      .= |((p1 <X> p2), p3)| by Th1;

      hence thesis;

    end;

    theorem :: EUCLID_8:95

     |{p1, p2, q}| = |((q <X> p1), p2)|

    proof

       |{p1, p2, q}| = |{p2, q, p1}| by Th84;

      hence thesis;

    end;

    begin

    definition

      let f1,f2,f3 be PartFunc of REAL , REAL ;

      let t0 be Real;

      :: EUCLID_8:def7

      func VFuncdiff (f1,f2,f3,t0) -> Element of ( REAL 3) equals |[( diff (f1,t0)), ( diff (f2,t0)), ( diff (f3,t0))]|;

      coherence ;

    end

    theorem :: EUCLID_8:96

    f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 implies ( VFuncdiff (f1,f2,f3,t0)) = (((( diff (f1,t0)) * <e1> ) + (( diff (f2,t0)) * <e2> )) + (( diff (f3,t0)) * <e3> )) by Th31;

    theorem :: EUCLID_8:97

    

     Th88: (f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0) & (g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0) implies ( VFuncdiff ((f1 + g1),(f2 + g2),(f3 + g3),t0)) = (( VFuncdiff (f1,f2,f3,t0)) + ( VFuncdiff (g1,g2,g3,t0)))

    proof

      assume that

       A1: f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 and

       A2: g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0;

      set p = |[( diff (f1,t0)), ( diff (f2,t0)), ( diff (f3,t0))]|;

      set q = |[( diff (g1,t0)), ( diff (g2,t0)), ( diff (g3,t0))]|;

      

       A3: (p . 1) = ( diff (f1,t0)) & (p . 2) = ( diff (f2,t0)) & (p . 3) = ( diff (f3,t0)) by FINSEQ_1: 45;

      

       A4: (q . 1) = ( diff (g1,t0)) & (q . 2) = ( diff (g2,t0)) & (q . 3) = ( diff (g3,t0)) by FINSEQ_1: 45;

      ( VFuncdiff ((f1 + g1),(f2 + g2),(f3 + g3),t0)) = |[(( diff (f1,t0)) + ( diff (g1,t0))), ( diff ((f2 + g2),t0)), ( diff ((f3 + g3),t0))]| by A1, A2, FDIFF_1: 13

      .= |[(( diff (f1,t0)) + ( diff (g1,t0))), (( diff (f2,t0)) + ( diff (g2,t0))), ( diff ((f3 + g3),t0))]| by A1, A2, FDIFF_1: 13

      .= |[((p . 1) + (q . 1)), ((p . 2) + (q . 2)), ((p . 3) + (q . 3))]| by A1, A2, A3, A4, FDIFF_1: 13

      .= (( VFuncdiff (f1,f2,f3,t0)) + ( VFuncdiff (g1,g2,g3,t0))) by Lm2;

      hence thesis;

    end;

    theorem :: EUCLID_8:98

    

     Th89: (f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0) & (g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0) implies ( VFuncdiff ((f1 - g1),(f2 - g2),(f3 - g3),t0)) = (( VFuncdiff (f1,f2,f3,t0)) - ( VFuncdiff (g1,g2,g3,t0)))

    proof

      assume that

       A1: f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 and

       A2: g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0;

      set p = |[( diff (f1,t0)), ( diff (f2,t0)), ( diff (f3,t0))]|;

      set q = |[( diff (g1,t0)), ( diff (g2,t0)), ( diff (g3,t0))]|;

      

       A3: (p . 1) = ( diff (f1,t0)) & (p . 2) = ( diff (f2,t0)) & (p . 3) = ( diff (f3,t0)) by FINSEQ_1: 45;

      

       A4: (q . 1) = ( diff (g1,t0)) & (q . 2) = ( diff (g2,t0)) & (q . 3) = ( diff (g3,t0)) by FINSEQ_1: 45;

      ( VFuncdiff ((f1 - g1),(f2 - g2),(f3 - g3),t0)) = |[(( diff (f1,t0)) - ( diff (g1,t0))), ( diff ((f2 - g2),t0)), ( diff ((f3 - g3),t0))]| by A1, A2, FDIFF_1: 14

      .= |[(( diff (f1,t0)) - ( diff (g1,t0))), (( diff (f2,t0)) - ( diff (g2,t0))), ( diff ((f3 - g3),t0))]| by A1, A2, FDIFF_1: 14

      .= |[((p . 1) - (q . 1)), ((p . 2) - (q . 2)), ((p . 3) - (q . 3))]| by A1, A2, A3, A4, FDIFF_1: 14

      .= (( VFuncdiff (f1,f2,f3,t0)) - ( VFuncdiff (g1,g2,g3,t0))) by Lm4;

      hence thesis;

    end;

    theorem :: EUCLID_8:99

    

     Th90: (f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0) implies ( VFuncdiff ((r (#) f1),(r (#) f2),(r (#) f3),t0)) = (r * ( VFuncdiff (f1,f2,f3,t0)))

    proof

      assume

       A1: f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0;

      set p = |[( diff (f1,t0)), ( diff (f2,t0)), ( diff (f3,t0))]|;

      

       A2: (p . 1) = ( diff (f1,t0)) & (p . 2) = ( diff (f2,t0)) & (p . 3) = ( diff (f3,t0)) by FINSEQ_1: 45;

      ( VFuncdiff ((r (#) f1),(r (#) f2),(r (#) f3),t0)) = |[(r * ( diff (f1,t0))), ( diff ((r (#) f2),t0)), ( diff ((r (#) f3),t0))]| by A1, FDIFF_1: 15

      .= |[(r * ( diff (f1,t0))), (r * ( diff (f2,t0))), ( diff ((r (#) f3),t0))]| by A1, FDIFF_1: 15

      .= |[(r * (p . 1)), (r * (p . 2)), (r * (p . 3))]| by A1, A2, FDIFF_1: 15

      .= (r * ( VFuncdiff (f1,f2,f3,t0))) by Lm1;

      hence thesis;

    end;

    theorem :: EUCLID_8:100

    

     Th91: (f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0) & (g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0) implies ( VFuncdiff ((f1 (#) g1),(f2 (#) g2),(f3 (#) g3),t0)) = ( |[((g1 . t0) * ( diff (f1,t0))), ((g2 . t0) * ( diff (f2,t0))), ((g3 . t0) * ( diff (f3,t0)))]| + |[((f1 . t0) * ( diff (g1,t0))), ((f2 . t0) * ( diff (g2,t0))), ((f3 . t0) * ( diff (g3,t0)))]|)

    proof

      assume that

       A1: f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 and

       A2: g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0;

      set p = |[((g1 . t0) * ( diff (f1,t0))), ((g2 . t0) * ( diff (f2,t0))), ((g3 . t0) * ( diff (f3,t0)))]|;

      set q = |[((f1 . t0) * ( diff (g1,t0))), ((f2 . t0) * ( diff (g2,t0))), ((f3 . t0) * ( diff (g3,t0)))]|;

      

       A3: (p . 1) = ((g1 . t0) * ( diff (f1,t0))) & (p . 2) = ((g2 . t0) * ( diff (f2,t0))) & (p . 3) = ((g3 . t0) * ( diff (f3,t0))) by FINSEQ_1: 45;

      

       A4: (q . 1) = ((f1 . t0) * ( diff (g1,t0))) & (q . 2) = ((f2 . t0) * ( diff (g2,t0))) & (q . 3) = ((f3 . t0) * ( diff (g3,t0))) by FINSEQ_1: 45;

      ( VFuncdiff ((f1 (#) g1),(f2 (#) g2),(f3 (#) g3),t0)) = |[(((g1 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g1,t0)))), ( diff ((f2 (#) g2),t0)), ( diff ((f3 (#) g3),t0))]| by A1, A2, FDIFF_1: 16

      .= |[(((g1 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g1,t0)))), (((g2 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g2,t0)))), ( diff ((f3 (#) g3),t0))]| by A1, A2, FDIFF_1: 16

      .= |[((p . 1) + (q . 1)), ((p . 2) + (q . 2)), ((p . 3) + (q . 3))]| by A1, A2, A3, A4, FDIFF_1: 16

      .= ( |[((g1 . t0) * ( diff (f1,t0))), ((g2 . t0) * ( diff (f2,t0))), ((g3 . t0) * ( diff (f3,t0)))]| + |[((f1 . t0) * ( diff (g1,t0))), ((f2 . t0) * ( diff (g2,t0))), ((f3 . t0) * ( diff (g3,t0)))]|) by Lm2;

      hence thesis;

    end;

    theorem :: EUCLID_8:101

    

     Th92: (f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0) & (g1 is_differentiable_in (f1 . t0) & g2 is_differentiable_in (f2 . t0) & g3 is_differentiable_in (f3 . t0)) implies ( VFuncdiff ((g1 * f1),(g2 * f2),(g3 * f3),t0)) = |[(( diff (g1,(f1 . t0))) * ( diff (f1,t0))), (( diff (g2,(f2 . t0))) * ( diff (f2,t0))), (( diff (g3,(f3 . t0))) * ( diff (f3,t0)))]|

    proof

      assume that

       A1: f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 and

       A2: g1 is_differentiable_in (f1 . t0) & g2 is_differentiable_in (f2 . t0) & g3 is_differentiable_in (f3 . t0);

      ( VFuncdiff ((g1 * f1),(g2 * f2),(g3 * f3),t0)) = |[(( diff (g1,(f1 . t0))) * ( diff (f1,t0))), ( diff ((g2 * f2),t0)), ( diff ((g3 * f3),t0))]| by A1, A2, FDIFF_2: 13

      .= |[(( diff (g1,(f1 . t0))) * ( diff (f1,t0))), (( diff (g2,(f2 . t0))) * ( diff (f2,t0))), ( diff ((g3 * f3),t0))]| by A1, A2, FDIFF_2: 13

      .= |[(( diff (g1,(f1 . t0))) * ( diff (f1,t0))), (( diff (g2,(f2 . t0))) * ( diff (f2,t0))), (( diff (g3,(f3 . t0))) * ( diff (f3,t0)))]| by A1, A2, FDIFF_2: 13;

      hence thesis;

    end;

    theorem :: EUCLID_8:102

    (f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0) & (g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0) & ((g1 . t0) <> 0 & (g2 . t0) <> 0 & (g3 . t0) <> 0 ) implies ( VFuncdiff ((f1 / g1),(f2 / g2),(f3 / g3),t0)) = |[(((( diff (f1,t0)) * (g1 . t0)) - (( diff (g1,t0)) * (f1 . t0))) / ((g1 . t0) ^2 )), (((( diff (f2,t0)) * (g2 . t0)) - (( diff (g2,t0)) * (f2 . t0))) / ((g2 . t0) ^2 )), (((( diff (f3,t0)) * (g3 . t0)) - (( diff (g3,t0)) * (f3 . t0))) / ((g3 . t0) ^2 ))]|

    proof

      assume that

       A1: f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 and

       A2: g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 and

       A3: (g1 . t0) <> 0 & (g2 . t0) <> 0 & (g3 . t0) <> 0 ;

      ( VFuncdiff ((f1 / g1),(f2 / g2),(f3 / g3),t0)) = |[(((( diff (f1,t0)) * (g1 . t0)) - (( diff (g1,t0)) * (f1 . t0))) / ((g1 . t0) ^2 )), ( diff ((f2 / g2),t0)), ( diff ((f3 / g3),t0))]| by A1, A2, A3, FDIFF_2: 14

      .= |[(((( diff (f1,t0)) * (g1 . t0)) - (( diff (g1,t0)) * (f1 . t0))) / ((g1 . t0) ^2 )), (((( diff (f2,t0)) * (g2 . t0)) - (( diff (g2,t0)) * (f2 . t0))) / ((g2 . t0) ^2 )), ( diff ((f3 / g3),t0))]| by A1, A2, A3, FDIFF_2: 14

      .= |[(((( diff (f1,t0)) * (g1 . t0)) - (( diff (g1,t0)) * (f1 . t0))) / ((g1 . t0) ^2 )), (((( diff (f2,t0)) * (g2 . t0)) - (( diff (g2,t0)) * (f2 . t0))) / ((g2 . t0) ^2 )), (((( diff (f3,t0)) * (g3 . t0)) - (( diff (g3,t0)) * (f3 . t0))) / ((g3 . t0) ^2 ))]| by A1, A2, A3, FDIFF_2: 14;

      hence thesis;

    end;

    theorem :: EUCLID_8:103

    

     Th94: (f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0) & ((f1 . t0) <> 0 & (f2 . t0) <> 0 & (f3 . t0) <> 0 ) implies ( VFuncdiff ((f1 ^ ),(f2 ^ ),(f3 ^ ),t0)) = ( - |[(( diff (f1,t0)) / ((f1 . t0) ^2 )), (( diff (f2,t0)) / ((f2 . t0) ^2 )), (( diff (f3,t0)) / ((f3 . t0) ^2 ))]|)

    proof

      assume that

       A1: f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 and

       A2: (f1 . t0) <> 0 & (f2 . t0) <> 0 & (f3 . t0) <> 0 ;

      set p = |[(( diff (f1,t0)) / ((f1 . t0) ^2 )), (( diff (f2,t0)) / ((f2 . t0) ^2 )), (( diff (f3,t0)) / ((f3 . t0) ^2 ))]|;

      

       A3: (p . 1) = (( diff (f1,t0)) / ((f1 . t0) ^2 )) & (p . 2) = (( diff (f2,t0)) / ((f2 . t0) ^2 )) & (p . 3) = (( diff (f3,t0)) / ((f3 . t0) ^2 )) by FINSEQ_1: 45;

      ( VFuncdiff ((f1 ^ ),(f2 ^ ),(f3 ^ ),t0)) = |[( - (( diff (f1,t0)) / ((f1 . t0) ^2 ))), ( diff ((f2 ^ ),t0)), ( diff ((f3 ^ ),t0))]| by A1, A2, FDIFF_2: 15

      .= |[( - (( diff (f1,t0)) / ((f1 . t0) ^2 ))), ( - (( diff (f2,t0)) / ((f2 . t0) ^2 ))), ( diff ((f3 ^ ),t0))]| by A1, A2, FDIFF_2: 15

      .= |[( - (( diff (f1,t0)) / ((f1 . t0) ^2 ))), ( - (( diff (f2,t0)) / ((f2 . t0) ^2 ))), ( - (( diff (f3,t0)) / ((f3 . t0) ^2 )))]| by A1, A2, FDIFF_2: 15

      .= |[(( - 1) * (p . 1)), (( - 1) * (p . 2)), (( - 1) * (p . 3))]| by A3

      .= ( - |[(( diff (f1,t0)) / ((f1 . t0) ^2 )), (( diff (f2,t0)) / ((f2 . t0) ^2 )), (( diff (f3,t0)) / ((f3 . t0) ^2 ))]|) by Lm1;

      hence thesis;

    end;

    theorem :: EUCLID_8:104

    f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 implies ( VFuncdiff ((r (#) f1),(r (#) f2),(r (#) f3),t0)) = ((((r * ( diff (f1,t0))) * <e1> ) + ((r * ( diff (f2,t0))) * <e2> )) + ((r * ( diff (f3,t0))) * <e3> ))

    proof

      assume f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0;

      

      then ( VFuncdiff ((r (#) f1),(r (#) f2),(r (#) f3),t0)) = (r * ( VFuncdiff (f1,f2,f3,t0))) by Th90

      .= (r * (((( diff (f1,t0)) * <e1> ) + (( diff (f2,t0)) * <e2> )) + (( diff (f3,t0)) * <e3> ))) by Th31

      .= (r * (( |[( diff (f1,t0)), 0 , 0 ]| + (( diff (f2,t0)) * <e2> )) + (( diff (f3,t0)) * <e3> ))) by Th21

      .= (r * (( |[( diff (f1,t0)), 0 , 0 ]| + |[ 0 , ( diff (f2,t0)), 0 ]|) + (( diff (f3,t0)) * <e3> ))) by Th22

      .= (r * (( |[( diff (f1,t0)), 0 , 0 ]| + |[ 0 , ( diff (f2,t0)), 0 ]|) + |[ 0 , 0 , ( diff (f3,t0))]|)) by Th23

      .= (r * ( |[(( diff (f1,t0)) + 0 ), ( 0 + ( diff (f2,t0))), ( 0 + 0 )]| + |[ 0 , 0 , ( diff (f3,t0))]|)) by Lm8

      .= (r * |[(( diff (f1,t0)) + 0 ), (( diff (f2,t0)) + 0 ), ( 0 + ( diff (f3,t0)))]|) by Lm8

      .= ((((r * ( diff (f1,t0))) * <e1> ) + ((r * ( diff (f2,t0))) * <e2> )) + ((r * ( diff (f3,t0))) * <e3> )) by Th32;

      hence thesis;

    end;

    theorem :: EUCLID_8:105

    (f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0) & (g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0) implies ( VFuncdiff ((r (#) (f1 + g1)),(r (#) (f2 + g2)),(r (#) (f3 + g3)),t0)) = ((r * ( VFuncdiff (f1,f2,f3,t0))) + (r * ( VFuncdiff (g1,g2,g3,t0))))

    proof

      assume that

       A1: f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 and

       A2: g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0;

      (f1 + g1) is_differentiable_in t0 & (f2 + g2) is_differentiable_in t0 & (f3 + g3) is_differentiable_in t0 by A1, A2, FDIFF_1: 13;

      

      then ( VFuncdiff ((r (#) (f1 + g1)),(r (#) (f2 + g2)),(r (#) (f3 + g3)),t0)) = (r * ( VFuncdiff ((f1 + g1),(f2 + g2),(f3 + g3),t0))) by Th90

      .= (r * (( VFuncdiff (f1,f2,f3,t0)) + ( VFuncdiff (g1,g2,g3,t0)))) by A1, A2, Th88

      .= ((r * ( VFuncdiff (f1,f2,f3,t0))) + (r * ( VFuncdiff (g1,g2,g3,t0)))) by EUCLID_4: 6;

      hence thesis;

    end;

    theorem :: EUCLID_8:106

    (f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0) & (g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0) implies ( VFuncdiff ((r (#) (f1 - g1)),(r (#) (f2 - g2)),(r (#) (f3 - g3)),t0)) = ((r * ( VFuncdiff (f1,f2,f3,t0))) - (r * ( VFuncdiff (g1,g2,g3,t0))))

    proof

      assume that

       A1: f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 and

       A2: g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0;

      (f1 - g1) is_differentiable_in t0 & (f2 - g2) is_differentiable_in t0 & (f3 - g3) is_differentiable_in t0 by A1, A2, FDIFF_1: 14;

      

      then ( VFuncdiff ((r (#) (f1 - g1)),(r (#) (f2 - g2)),(r (#) (f3 - g3)),t0)) = (r * ( VFuncdiff ((f1 - g1),(f2 - g2),(f3 - g3),t0))) by Th90

      .= (r * (( VFuncdiff (f1,f2,f3,t0)) - ( VFuncdiff (g1,g2,g3,t0)))) by A1, A2, Th89

      .= ((r * ( VFuncdiff (f1,f2,f3,t0))) - (r * ( VFuncdiff (g1,g2,g3,t0)))) by EUCLIDLP: 12;

      hence thesis;

    end;

    theorem :: EUCLID_8:107

    (f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0) & (g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0) implies ( VFuncdiff (((r (#) f1) (#) g1),((r (#) f2) (#) g2),((r (#) f3) (#) g3),t0)) = ((r * |[((g1 . t0) * ( diff (f1,t0))), ((g2 . t0) * ( diff (f2,t0))), ((g3 . t0) * ( diff (f3,t0)))]|) + (r * |[((f1 . t0) * ( diff (g1,t0))), ((f2 . t0) * ( diff (g2,t0))), ((f3 . t0) * ( diff (g3,t0)))]|))

    proof

      assume that

       A1: f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 and

       A2: g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0;

      (r (#) f1) is_differentiable_in t0 & (r (#) f2) is_differentiable_in t0 & (r (#) f3) is_differentiable_in t0 by A1, FDIFF_1: 15;

      

      then ( VFuncdiff (((r (#) f1) (#) g1),((r (#) f2) (#) g2),((r (#) f3) (#) g3),t0)) = ( |[((g1 . t0) * ( diff ((r (#) f1),t0))), ((g2 . t0) * ( diff ((r (#) f2),t0))), ((g3 . t0) * ( diff ((r (#) f3),t0)))]| + |[(((r (#) f1) . t0) * ( diff (g1,t0))), (((r (#) f2) . t0) * ( diff (g2,t0))), (((r (#) f3) . t0) * ( diff (g3,t0)))]|) by A2, Th91

      .= ( |[((g1 . t0) * (r * ( diff (f1,t0)))), ((g2 . t0) * ( diff ((r (#) f2),t0))), ((g3 . t0) * ( diff ((r (#) f3),t0)))]| + |[(((r (#) f1) . t0) * ( diff (g1,t0))), (((r (#) f2) . t0) * ( diff (g2,t0))), (((r (#) f3) . t0) * ( diff (g3,t0)))]|) by A1, FDIFF_1: 15

      .= ( |[((g1 . t0) * (r * ( diff (f1,t0)))), ((g2 . t0) * (r * ( diff (f2,t0)))), ((g3 . t0) * ( diff ((r (#) f3),t0)))]| + |[(((r (#) f1) . t0) * ( diff (g1,t0))), (((r (#) f2) . t0) * ( diff (g2,t0))), (((r (#) f3) . t0) * ( diff (g3,t0)))]|) by A1, FDIFF_1: 15

      .= ( |[((g1 . t0) * (r * ( diff (f1,t0)))), ((g2 . t0) * (r * ( diff (f2,t0)))), ((g3 . t0) * (r * ( diff (f3,t0))))]| + |[(((r (#) f1) . t0) * ( diff (g1,t0))), (((r (#) f2) . t0) * ( diff (g2,t0))), (((r (#) f3) . t0) * ( diff (g3,t0)))]|) by A1, FDIFF_1: 15

      .= ( |[((g1 . t0) * (r * ( diff (f1,t0)))), ((g2 . t0) * (r * ( diff (f2,t0)))), ((g3 . t0) * (r * ( diff (f3,t0))))]| + |[((r * (f1 . t0)) * ( diff (g1,t0))), (((r (#) f2) . t0) * ( diff (g2,t0))), (((r (#) f3) . t0) * ( diff (g3,t0)))]|) by VALUED_1: 6

      .= ( |[((g1 . t0) * (r * ( diff (f1,t0)))), ((g2 . t0) * (r * ( diff (f2,t0)))), ((g3 . t0) * (r * ( diff (f3,t0))))]| + |[((r * (f1 . t0)) * ( diff (g1,t0))), ((r * (f2 . t0)) * ( diff (g2,t0))), (((r (#) f3) . t0) * ( diff (g3,t0)))]|) by VALUED_1: 6

      .= ( |[((g1 . t0) * (r * ( diff (f1,t0)))), ((g2 . t0) * (r * ( diff (f2,t0)))), ((g3 . t0) * (r * ( diff (f3,t0))))]| + |[((r * (f1 . t0)) * ( diff (g1,t0))), ((r * (f2 . t0)) * ( diff (g2,t0))), ((r * (f3 . t0)) * ( diff (g3,t0)))]|) by VALUED_1: 6

      .= ( |[((g1 . t0) * (r * ( diff (f1,t0)))), ((g2 . t0) * (r * ( diff (f2,t0)))), ((g3 . t0) * (r * ( diff (f3,t0))))]| + |[(r * ((f1 . t0) * ( diff (g1,t0)))), (r * ((f2 . t0) * ( diff (g2,t0)))), (r * ((f3 . t0) * ( diff (g3,t0))))]|)

      .= ( |[(r * ((g1 . t0) * ( diff (f1,t0)))), (r * ((g2 . t0) * ( diff (f2,t0)))), (r * ((g3 . t0) * ( diff (f3,t0))))]| + (r * |[((f1 . t0) * ( diff (g1,t0))), ((f2 . t0) * ( diff (g2,t0))), ((f3 . t0) * ( diff (g3,t0)))]|)) by Lm6

      .= ((r * |[((g1 . t0) * ( diff (f1,t0))), ((g2 . t0) * ( diff (f2,t0))), ((g3 . t0) * ( diff (f3,t0)))]|) + (r * |[((f1 . t0) * ( diff (g1,t0))), ((f2 . t0) * ( diff (g2,t0))), ((f3 . t0) * ( diff (g3,t0)))]|)) by Lm6;

      hence thesis;

    end;

    theorem :: EUCLID_8:108

    (f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0) & (g1 is_differentiable_in (f1 . t0) & g2 is_differentiable_in (f2 . t0) & g3 is_differentiable_in (f3 . t0)) implies ( VFuncdiff (((r (#) g1) * f1),((r (#) g2) * f2),((r (#) g3) * f3),t0)) = (r * |[(( diff (g1,(f1 . t0))) * ( diff (f1,t0))), (( diff (g2,(f2 . t0))) * ( diff (f2,t0))), (( diff (g3,(f3 . t0))) * ( diff (f3,t0)))]|)

    proof

      assume that

       A1: f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 and

       A2: g1 is_differentiable_in (f1 . t0) & g2 is_differentiable_in (f2 . t0) & g3 is_differentiable_in (f3 . t0);

      (r (#) g1) is_differentiable_in (f1 . t0) & (r (#) g2) is_differentiable_in (f2 . t0) & (r (#) g3) is_differentiable_in (f3 . t0) by A2, FDIFF_1: 15;

      

      then ( VFuncdiff (((r (#) g1) * f1),((r (#) g2) * f2),((r (#) g3) * f3),t0)) = |[(( diff ((r (#) g1),(f1 . t0))) * ( diff (f1,t0))), (( diff ((r (#) g2),(f2 . t0))) * ( diff (f2,t0))), (( diff ((r (#) g3),(f3 . t0))) * ( diff (f3,t0)))]| by A1, Th92

      .= |[((r * ( diff (g1,(f1 . t0)))) * ( diff (f1,t0))), (( diff ((r (#) g2),(f2 . t0))) * ( diff (f2,t0))), (( diff ((r (#) g3),(f3 . t0))) * ( diff (f3,t0)))]| by A2, FDIFF_1: 15

      .= |[((r * ( diff (g1,(f1 . t0)))) * ( diff (f1,t0))), ((r * ( diff (g2,(f2 . t0)))) * ( diff (f2,t0))), (( diff ((r (#) g3),(f3 . t0))) * ( diff (f3,t0)))]| by A2, FDIFF_1: 15

      .= |[((r * ( diff (g1,(f1 . t0)))) * ( diff (f1,t0))), ((r * ( diff (g2,(f2 . t0)))) * ( diff (f2,t0))), ((r * ( diff (g3,(f3 . t0)))) * ( diff (f3,t0)))]| by A2, FDIFF_1: 15

      .= |[(r * (( diff (g1,(f1 . t0))) * ( diff (f1,t0)))), (r * (( diff (g2,(f2 . t0))) * ( diff (f2,t0)))), (r * (( diff (g3,(f3 . t0))) * ( diff (f3,t0))))]|

      .= (r * |[(( diff (g1,(f1 . t0))) * ( diff (f1,t0))), (( diff (g2,(f2 . t0))) * ( diff (f2,t0))), (( diff (g3,(f3 . t0))) * ( diff (f3,t0)))]|) by Lm6;

      hence thesis;

    end;

    theorem :: EUCLID_8:109

    (f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0) & (g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0) & ((g1 . t0) <> 0 & (g2 . t0) <> 0 & (g3 . t0) <> 0 ) implies ( VFuncdiff (((r (#) f1) / g1),((r (#) f2) / g2),((r (#) f3) / g3),t0)) = (r * |[(((( diff (f1,t0)) * (g1 . t0)) - (( diff (g1,t0)) * (f1 . t0))) / ((g1 . t0) ^2 )), (((( diff (f2,t0)) * (g2 . t0)) - (( diff (g2,t0)) * (f2 . t0))) / ((g2 . t0) ^2 )), (((( diff (f3,t0)) * (g3 . t0)) - (( diff (g3,t0)) * (f3 . t0))) / ((g3 . t0) ^2 ))]|)

    proof

      assume that

       A1: f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 and

       A2: g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 and

       A3: (g1 . t0) <> 0 & (g2 . t0) <> 0 & (g3 . t0) <> 0 ;

      

       A4: (r (#) f1) is_differentiable_in t0 & (r (#) f2) is_differentiable_in t0 & (r (#) f3) is_differentiable_in t0 by A1, FDIFF_1: 15;

      

      then ( VFuncdiff (((r (#) f1) / g1),((r (#) f2) / g2),((r (#) f3) / g3),t0)) = |[(((( diff ((r (#) f1),t0)) * (g1 . t0)) - (( diff (g1,t0)) * ((r (#) f1) . t0))) / ((g1 . t0) ^2 )), ( diff (((r (#) f2) / g2),t0)), ( diff (((r (#) f3) / g3),t0))]| by A2, A3, FDIFF_2: 14

      .= |[(((( diff ((r (#) f1),t0)) * (g1 . t0)) - (( diff (g1,t0)) * ((r (#) f1) . t0))) / ((g1 . t0) ^2 )), (((( diff ((r (#) f2),t0)) * (g2 . t0)) - (( diff (g2,t0)) * ((r (#) f2) . t0))) / ((g2 . t0) ^2 )), ( diff (((r (#) f3) / g3),t0))]| by A2, A3, A4, FDIFF_2: 14

      .= |[(((( diff ((r (#) f1),t0)) * (g1 . t0)) - (( diff (g1,t0)) * ((r (#) f1) . t0))) / ((g1 . t0) ^2 )), (((( diff ((r (#) f2),t0)) * (g2 . t0)) - (( diff (g2,t0)) * ((r (#) f2) . t0))) / ((g2 . t0) ^2 )), (((( diff ((r (#) f3),t0)) * (g3 . t0)) - (( diff (g3,t0)) * ((r (#) f3) . t0))) / ((g3 . t0) ^2 ))]| by A2, A3, A4, FDIFF_2: 14

      .= |[((((r * ( diff (f1,t0))) * (g1 . t0)) - (( diff (g1,t0)) * ((r (#) f1) . t0))) / ((g1 . t0) ^2 )), (((( diff ((r (#) f2),t0)) * (g2 . t0)) - (( diff (g2,t0)) * ((r (#) f2) . t0))) / ((g2 . t0) ^2 )), (((( diff ((r (#) f3),t0)) * (g3 . t0)) - (( diff (g3,t0)) * ((r (#) f3) . t0))) / ((g3 . t0) ^2 ))]| by A1, FDIFF_1: 15

      .= |[((((r * ( diff (f1,t0))) * (g1 . t0)) - (( diff (g1,t0)) * ((r (#) f1) . t0))) / ((g1 . t0) ^2 )), ((((r * ( diff (f2,t0))) * (g2 . t0)) - (( diff (g2,t0)) * ((r (#) f2) . t0))) / ((g2 . t0) ^2 )), (((( diff ((r (#) f3),t0)) * (g3 . t0)) - (( diff (g3,t0)) * ((r (#) f3) . t0))) / ((g3 . t0) ^2 ))]| by A1, FDIFF_1: 15

      .= |[((((r * ( diff (f1,t0))) * (g1 . t0)) - (( diff (g1,t0)) * ((r (#) f1) . t0))) / ((g1 . t0) ^2 )), ((((r * ( diff (f2,t0))) * (g2 . t0)) - (( diff (g2,t0)) * ((r (#) f2) . t0))) / ((g2 . t0) ^2 )), ((((r * ( diff (f3,t0))) * (g3 . t0)) - (( diff (g3,t0)) * ((r (#) f3) . t0))) / ((g3 . t0) ^2 ))]| by A1, FDIFF_1: 15

      .= |[((((r * ( diff (f1,t0))) * (g1 . t0)) - (( diff (g1,t0)) * (r * (f1 . t0)))) / ((g1 . t0) ^2 )), ((((r * ( diff (f2,t0))) * (g2 . t0)) - (( diff (g2,t0)) * ((r (#) f2) . t0))) / ((g2 . t0) ^2 )), ((((r * ( diff (f3,t0))) * (g3 . t0)) - (( diff (g3,t0)) * ((r (#) f3) . t0))) / ((g3 . t0) ^2 ))]| by VALUED_1: 6

      .= |[((((r * ( diff (f1,t0))) * (g1 . t0)) - (( diff (g1,t0)) * (r * (f1 . t0)))) / ((g1 . t0) ^2 )), ((((r * ( diff (f2,t0))) * (g2 . t0)) - (( diff (g2,t0)) * (r * (f2 . t0)))) / ((g2 . t0) ^2 )), ((((r * ( diff (f3,t0))) * (g3 . t0)) - (( diff (g3,t0)) * ((r (#) f3) . t0))) / ((g3 . t0) ^2 ))]| by VALUED_1: 6

      .= |[((((r * ( diff (f1,t0))) * (g1 . t0)) - (( diff (g1,t0)) * (r * (f1 . t0)))) / ((g1 . t0) ^2 )), ((((r * ( diff (f2,t0))) * (g2 . t0)) - (( diff (g2,t0)) * (r * (f2 . t0)))) / ((g2 . t0) ^2 )), ((((r * ( diff (f3,t0))) * (g3 . t0)) - (( diff (g3,t0)) * (r * (f3 . t0)))) / ((g3 . t0) ^2 ))]| by VALUED_1: 6

      .= |[((r * ((( diff (f1,t0)) * (g1 . t0)) - (( diff (g1,t0)) * (f1 . t0)))) / ((g1 . t0) ^2 )), ((r * ((( diff (f2,t0)) * (g2 . t0)) - (( diff (g2,t0)) * (f2 . t0)))) / ((g2 . t0) ^2 )), ((r * ((( diff (f3,t0)) * (g3 . t0)) - (( diff (g3,t0)) * (f3 . t0)))) / ((g3 . t0) ^2 ))]|

      .= |[(r * (((( diff (f1,t0)) * (g1 . t0)) - (( diff (g1,t0)) * (f1 . t0))) / ((g1 . t0) ^2 ))), ((r * ((( diff (f2,t0)) * (g2 . t0)) - (( diff (g2,t0)) * (f2 . t0)))) / ((g2 . t0) ^2 )), ((r * ((( diff (f3,t0)) * (g3 . t0)) - (( diff (g3,t0)) * (f3 . t0)))) / ((g3 . t0) ^2 ))]| by XCMPLX_1: 74

      .= |[(r * (((( diff (f1,t0)) * (g1 . t0)) - (( diff (g1,t0)) * (f1 . t0))) / ((g1 . t0) ^2 ))), (r * (((( diff (f2,t0)) * (g2 . t0)) - (( diff (g2,t0)) * (f2 . t0))) / ((g2 . t0) ^2 ))), ((r * ((( diff (f3,t0)) * (g3 . t0)) - (( diff (g3,t0)) * (f3 . t0)))) / ((g3 . t0) ^2 ))]| by XCMPLX_1: 74

      .= |[(r * (((( diff (f1,t0)) * (g1 . t0)) - (( diff (g1,t0)) * (f1 . t0))) / ((g1 . t0) ^2 ))), (r * (((( diff (f2,t0)) * (g2 . t0)) - (( diff (g2,t0)) * (f2 . t0))) / ((g2 . t0) ^2 ))), (r * (((( diff (f3,t0)) * (g3 . t0)) - (( diff (g3,t0)) * (f3 . t0))) / ((g3 . t0) ^2 )))]| by XCMPLX_1: 74

      .= (r * |[(((( diff (f1,t0)) * (g1 . t0)) - (( diff (g1,t0)) * (f1 . t0))) / ((g1 . t0) ^2 )), (((( diff (f2,t0)) * (g2 . t0)) - (( diff (g2,t0)) * (f2 . t0))) / ((g2 . t0) ^2 )), (((( diff (f3,t0)) * (g3 . t0)) - (( diff (g3,t0)) * (f3 . t0))) / ((g3 . t0) ^2 ))]|) by Lm6;

      hence thesis;

    end;

    theorem :: EUCLID_8:110

    (f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0) & ((f1 . t0) <> 0 & (f2 . t0) <> 0 & (f3 . t0) <> 0 ) & r <> 0 implies ( VFuncdiff (((r (#) f1) ^ ),((r (#) f2) ^ ),((r (#) f3) ^ ),t0)) = ( - ((1 / r) * |[(( diff (f1,t0)) / ((f1 . t0) ^2 )), (( diff (f2,t0)) / ((f2 . t0) ^2 )), (( diff (f3,t0)) / ((f3 . t0) ^2 ))]|))

    proof

      assume that

       A1: f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 and

       A2: (f1 . t0) <> 0 & (f2 . t0) <> 0 & (f3 . t0) <> 0 and

       A3: r <> 0 ;

      

       A4: (r (#) f1) is_differentiable_in t0 & (r (#) f2) is_differentiable_in t0 & (r (#) f3) is_differentiable_in t0 by A1, FDIFF_1: 15;

      

       A5: ((r (#) f1) . t0) = (r * (f1 . t0)) by VALUED_1: 6;

      

       A6: ((r (#) f2) . t0) = (r * (f2 . t0)) by VALUED_1: 6;

      

       A7: ((r (#) f3) . t0) = (r * (f3 . t0)) by VALUED_1: 6;

      

      then ( VFuncdiff (((r (#) f1) ^ ),((r (#) f2) ^ ),((r (#) f3) ^ ),t0)) = ( - |[(( diff ((r (#) f1),t0)) / (((r (#) f1) . t0) ^2 )), (( diff ((r (#) f2),t0)) / (((r (#) f2) . t0) ^2 )), (( diff ((r (#) f3),t0)) / (((r (#) f3) . t0) ^2 ))]|) by A4, A5, A6, A2, A3, Th94

      .= ( - |[((r * ( diff (f1,t0))) / (((r (#) f1) . t0) ^2 )), (( diff ((r (#) f2),t0)) / (((r (#) f2) . t0) ^2 )), (( diff ((r (#) f3),t0)) / (((r (#) f3) . t0) ^2 ))]|) by A1, FDIFF_1: 15

      .= ( - |[((r * ( diff (f1,t0))) / (((r (#) f1) . t0) ^2 )), ((r * ( diff (f2,t0))) / (((r (#) f2) . t0) ^2 )), (( diff ((r (#) f3),t0)) / (((r (#) f3) . t0) ^2 ))]|) by A1, FDIFF_1: 15

      .= ( - |[((r * ( diff (f1,t0))) / (((r (#) f1) . t0) ^2 )), ((r * ( diff (f2,t0))) / (((r (#) f2) . t0) ^2 )), ((r * ( diff (f3,t0))) / (((r (#) f3) . t0) ^2 ))]|) by A1, FDIFF_1: 15

      .= ( - |[(r * (( diff (f1,t0)) / (((r (#) f1) . t0) ^2 ))), ((r * ( diff (f2,t0))) / (((r (#) f2) . t0) ^2 )), ((r * ( diff (f3,t0))) / (((r (#) f3) . t0) ^2 ))]|) by XCMPLX_1: 74

      .= ( - |[(r * (( diff (f1,t0)) / (((r (#) f1) . t0) ^2 ))), (r * (( diff (f2,t0)) / (((r (#) f2) . t0) ^2 ))), ((r * ( diff (f3,t0))) / (((r (#) f3) . t0) ^2 ))]|) by XCMPLX_1: 74

      .= ( - |[(r * (( diff (f1,t0)) / (((r (#) f1) . t0) ^2 ))), (r * (( diff (f2,t0)) / (((r (#) f2) . t0) ^2 ))), (r * (( diff (f3,t0)) / (((r (#) f3) . t0) ^2 )))]|) by XCMPLX_1: 74

      .= ( - (r * |[(( diff (f1,t0)) / ((r ^2 ) * ((f1 . t0) ^2 ))), (( diff (f2,t0)) / ((r ^2 ) * ((f2 . t0) ^2 ))), (( diff (f3,t0)) / ((r ^2 ) * ((f3 . t0) ^2 )))]|)) by A7, A6, A5, Lm6

      .= ( - (r * |[((( diff (f1,t0)) / (r ^2 )) / ((f1 . t0) ^2 )), (( diff (f2,t0)) / ((r ^2 ) * ((f2 . t0) ^2 ))), (( diff (f3,t0)) / ((r ^2 ) * ((f3 . t0) ^2 )))]|)) by XCMPLX_1: 78

      .= ( - (r * |[((( diff (f1,t0)) / (r ^2 )) / ((f1 . t0) ^2 )), ((( diff (f2,t0)) / (r ^2 )) / ((f2 . t0) ^2 )), (( diff (f3,t0)) / ((r ^2 ) * ((f3 . t0) ^2 )))]|)) by XCMPLX_1: 78

      .= ( - (r * |[((( diff (f1,t0)) / (r ^2 )) / ((f1 . t0) ^2 )), ((( diff (f2,t0)) / (r ^2 )) / ((f2 . t0) ^2 )), ((( diff (f3,t0)) / (r ^2 )) / ((f3 . t0) ^2 ))]|)) by XCMPLX_1: 78

      .= ( - (r * |[((( diff (f1,t0)) / ((f1 . t0) ^2 )) / (r ^2 )), ((( diff (f2,t0)) / (r ^2 )) / ((f2 . t0) ^2 )), ((( diff (f3,t0)) / (r ^2 )) / ((f3 . t0) ^2 ))]|)) by XCMPLX_1: 48

      .= ( - (r * |[((( diff (f1,t0)) / ((f1 . t0) ^2 )) / (r ^2 )), ((( diff (f2,t0)) / ((f2 . t0) ^2 )) / (r ^2 )), ((( diff (f3,t0)) / (r ^2 )) / ((f3 . t0) ^2 ))]|)) by XCMPLX_1: 48

      .= ( - (r * |[((( diff (f1,t0)) / ((f1 . t0) ^2 )) / (r ^2 )), ((( diff (f2,t0)) / ((f2 . t0) ^2 )) / (r ^2 )), ((( diff (f3,t0)) / ((f3 . t0) ^2 )) / (r ^2 ))]|)) by XCMPLX_1: 48

      .= ( - (r * |[((( diff (f1,t0)) / ((f1 . t0) ^2 )) / ((1 / (r ^2 )) " )), ((( diff (f2,t0)) / ((f2 . t0) ^2 )) / (r ^2 )), ((( diff (f3,t0)) / ((f3 . t0) ^2 )) / (r ^2 ))]|)) by XCMPLX_1: 217

      .= ( - (r * |[((1 / (r ^2 )) * (( diff (f1,t0)) / ((f1 . t0) ^2 ))), ((( diff (f2,t0)) / ((f2 . t0) ^2 )) / (r ^2 )), ((( diff (f3,t0)) / ((f3 . t0) ^2 )) / (r ^2 ))]|)) by XCMPLX_1: 219

      .= ( - (r * |[((1 / (r ^2 )) * (( diff (f1,t0)) / ((f1 . t0) ^2 ))), ((( diff (f2,t0)) / ((f2 . t0) ^2 )) / ((1 / (r ^2 )) " )), ((( diff (f3,t0)) / ((f3 . t0) ^2 )) / (r ^2 ))]|)) by XCMPLX_1: 217

      .= ( - (r * |[((1 / (r ^2 )) * (( diff (f1,t0)) / ((f1 . t0) ^2 ))), ((1 / (r ^2 )) * (( diff (f2,t0)) / ((f2 . t0) ^2 ))), ((( diff (f3,t0)) / ((f3 . t0) ^2 )) / (r ^2 ))]|)) by XCMPLX_1: 219

      .= ( - (r * |[((1 / (r ^2 )) * (( diff (f1,t0)) / ((f1 . t0) ^2 ))), ((1 / (r ^2 )) * (( diff (f2,t0)) / ((f2 . t0) ^2 ))), ((( diff (f3,t0)) / ((f3 . t0) ^2 )) / ((1 / (r ^2 )) " ))]|)) by XCMPLX_1: 217

      .= ( - (r * |[((1 / (r ^2 )) * (( diff (f1,t0)) / ((f1 . t0) ^2 ))), ((1 / (r ^2 )) * (( diff (f2,t0)) / ((f2 . t0) ^2 ))), ((1 / (r ^2 )) * (( diff (f3,t0)) / ((f3 . t0) ^2 )))]|)) by XCMPLX_1: 219

      .= ( - (r * ((1 / (r ^2 )) * |[(( diff (f1,t0)) / ((f1 . t0) ^2 )), (( diff (f2,t0)) / ((f2 . t0) ^2 )), (( diff (f3,t0)) / ((f3 . t0) ^2 ))]|))) by Lm6

      .= ( - ((r * (1 / (r ^2 ))) * |[(( diff (f1,t0)) / ((f1 . t0) ^2 )), (( diff (f2,t0)) / ((f2 . t0) ^2 )), (( diff (f3,t0)) / ((f3 . t0) ^2 ))]|)) by EUCLID_4: 4

      .= ( - (((r * 1) / (r ^2 )) * |[(( diff (f1,t0)) / ((f1 . t0) ^2 )), (( diff (f2,t0)) / ((f2 . t0) ^2 )), (( diff (f3,t0)) / ((f3 . t0) ^2 ))]|)) by XCMPLX_1: 74

      .= ( - (((r / r) / r) * |[(( diff (f1,t0)) / ((f1 . t0) ^2 )), (( diff (f2,t0)) / ((f2 . t0) ^2 )), (( diff (f3,t0)) / ((f3 . t0) ^2 ))]|)) by XCMPLX_1: 78

      .= ( - ((1 / r) * |[(( diff (f1,t0)) / ((f1 . t0) ^2 )), (( diff (f2,t0)) / ((f2 . t0) ^2 )), (( diff (f3,t0)) / ((f3 . t0) ^2 ))]|)) by A3, XCMPLX_1: 60;

      hence thesis;

    end;

    theorem :: EUCLID_8:111

    (f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0) & (g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0) implies ( VFuncdiff (((f2 (#) g3) - (f3 (#) g2)),((f3 (#) g1) - (f1 (#) g3)),((f1 (#) g2) - (f2 (#) g1)),t0)) = ( |[(((f2 . t0) * ( diff (g3,t0))) - ((f3 . t0) * ( diff (g2,t0)))), (((f3 . t0) * ( diff (g1,t0))) - ((f1 . t0) * ( diff (g3,t0)))), (((f1 . t0) * ( diff (g2,t0))) - ((f2 . t0) * ( diff (g1,t0))))]| + |[((( diff (f2,t0)) * (g3 . t0)) - (( diff (f3,t0)) * (g2 . t0))), ((( diff (f3,t0)) * (g1 . t0)) - (( diff (f1,t0)) * (g3 . t0))), ((( diff (f1,t0)) * (g2 . t0)) - (( diff (f2,t0)) * (g1 . t0)))]|)

    proof

      assume that

       A1: f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 and

       A2: g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0;

      (f2 (#) g3) is_differentiable_in t0 & (f3 (#) g2) is_differentiable_in t0 & (f3 (#) g1) is_differentiable_in t0 & (f1 (#) g3) is_differentiable_in t0 & (f1 (#) g2) is_differentiable_in t0 & (f2 (#) g1) is_differentiable_in t0 by A1, A2, FDIFF_1: 16;

      

      then ( VFuncdiff (((f2 (#) g3) - (f3 (#) g2)),((f3 (#) g1) - (f1 (#) g3)),((f1 (#) g2) - (f2 (#) g1)),t0)) = (( VFuncdiff ((f2 (#) g3),(f3 (#) g1),(f1 (#) g2),t0)) - ( VFuncdiff ((f3 (#) g2),(f1 (#) g3),(f2 (#) g1),t0))) by Th89

      .= (( |[((g3 . t0) * ( diff (f2,t0))), ((g1 . t0) * ( diff (f3,t0))), ((g2 . t0) * ( diff (f1,t0)))]| + |[((f2 . t0) * ( diff (g3,t0))), ((f3 . t0) * ( diff (g1,t0))), ((f1 . t0) * ( diff (g2,t0)))]|) - ( VFuncdiff ((f3 (#) g2),(f1 (#) g3),(f2 (#) g1),t0))) by A1, A2, Th91

      .= (( |[((g3 . t0) * ( diff (f2,t0))), ((g1 . t0) * ( diff (f3,t0))), ((g2 . t0) * ( diff (f1,t0)))]| + |[((f2 . t0) * ( diff (g3,t0))), ((f3 . t0) * ( diff (g1,t0))), ((f1 . t0) * ( diff (g2,t0)))]|) - ( |[((g2 . t0) * ( diff (f3,t0))), ((g3 . t0) * ( diff (f1,t0))), ((g1 . t0) * ( diff (f2,t0)))]| + |[((f3 . t0) * ( diff (g2,t0))), ((f1 . t0) * ( diff (g3,t0))), ((f2 . t0) * ( diff (g1,t0)))]|)) by A1, A2, Th91

      .= (( |[((g3 . t0) * ( diff (f2,t0))), ((g1 . t0) * ( diff (f3,t0))), ((g2 . t0) * ( diff (f1,t0)))]| - |[((g2 . t0) * ( diff (f3,t0))), ((g3 . t0) * ( diff (f1,t0))), ((g1 . t0) * ( diff (f2,t0)))]|) + ( |[((f2 . t0) * ( diff (g3,t0))), ((f3 . t0) * ( diff (g1,t0))), ((f1 . t0) * ( diff (g2,t0)))]| - |[((f3 . t0) * ( diff (g2,t0))), ((f1 . t0) * ( diff (g3,t0))), ((f2 . t0) * ( diff (g1,t0)))]|)) by Lm10

      .= ( |[(((g3 . t0) * ( diff (f2,t0))) - ((g2 . t0) * ( diff (f3,t0)))), (((g1 . t0) * ( diff (f3,t0))) - ((g3 . t0) * ( diff (f1,t0)))), (((g2 . t0) * ( diff (f1,t0))) - ((g1 . t0) * ( diff (f2,t0))))]| + ( |[((f2 . t0) * ( diff (g3,t0))), ((f3 . t0) * ( diff (g1,t0))), ((f1 . t0) * ( diff (g2,t0)))]| - |[((f3 . t0) * ( diff (g2,t0))), ((f1 . t0) * ( diff (g3,t0))), ((f2 . t0) * ( diff (g1,t0)))]|)) by Lm11

      .= ( |[(((g3 . t0) * ( diff (f2,t0))) - ((g2 . t0) * ( diff (f3,t0)))), (((g1 . t0) * ( diff (f3,t0))) - ((g3 . t0) * ( diff (f1,t0)))), (((g2 . t0) * ( diff (f1,t0))) - ((g1 . t0) * ( diff (f2,t0))))]| + |[(((f2 . t0) * ( diff (g3,t0))) - ((f3 . t0) * ( diff (g2,t0)))), (((f3 . t0) * ( diff (g1,t0))) - ((f1 . t0) * ( diff (g3,t0)))), (((f1 . t0) * ( diff (g2,t0))) - ((f2 . t0) * ( diff (g1,t0))))]|) by Lm11;

      hence thesis;

    end;

    theorem :: EUCLID_8:112

    (f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0) & (g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0) & (h1 is_differentiable_in t0 & h2 is_differentiable_in t0 & h3 is_differentiable_in t0) implies ( VFuncdiff ((h1 (#) ((f2 (#) g3) - (f3 (#) g2))),(h2 (#) ((f3 (#) g1) - (f1 (#) g3))),(h3 (#) ((f1 (#) g2) - (f2 (#) g1))),t0)) = (( |[(( diff (h1,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))), (( diff (h2,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0)))), (( diff (h3,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))))]| + |[((h1 . t0) * ((( diff (f2,t0)) * (g3 . t0)) - (( diff (f3,t0)) * (g2 . t0)))), ((h2 . t0) * ((( diff (f3,t0)) * (g1 . t0)) - (( diff (f1,t0)) * (g3 . t0)))), ((h3 . t0) * ((( diff (f1,t0)) * (g2 . t0)) - (( diff (f2,t0)) * (g1 . t0))))]|) + |[((h1 . t0) * (((f2 . t0) * ( diff (g3,t0))) - ((f3 . t0) * ( diff (g2,t0))))), ((h2 . t0) * (((f3 . t0) * ( diff (g1,t0))) - ((f1 . t0) * ( diff (g3,t0))))), ((h3 . t0) * (((f1 . t0) * ( diff (g2,t0))) - ((f2 . t0) * ( diff (g1,t0)))))]|)

    proof

      assume that

       A1: f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 and

       A2: g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 and

       A3: h1 is_differentiable_in t0 & h2 is_differentiable_in t0 & h3 is_differentiable_in t0;

      

       A4: (f2 (#) g3) is_differentiable_in t0 & (f3 (#) g2) is_differentiable_in t0 & (f3 (#) g1) is_differentiable_in t0 & (f1 (#) g3) is_differentiable_in t0 & (f1 (#) g2) is_differentiable_in t0 & (f2 (#) g1) is_differentiable_in t0 by A1, A2, FDIFF_1: 16;

      then

       A5: ((f2 (#) g3) - (f3 (#) g2)) is_differentiable_in t0 & ((f3 (#) g1) - (f1 (#) g3)) is_differentiable_in t0 & ((f1 (#) g2) - (f2 (#) g1)) is_differentiable_in t0 by FDIFF_1: 14;

      then

       A6: ((f2 (#) g3) - (f3 (#) g2)) is_left_differentiable_in t0 by FDIFF_3: 22;

      

       A7: t0 in ( dom ((f2 (#) g3) - (f3 (#) g2)))

      proof

        consider r such that

         A8: 0 < r & [.(t0 - r), t0.] c= ( dom ((f2 (#) g3) - (f3 (#) g2))) by A6, FDIFF_3:def 4;

        (t0 - r) <= t0 by A8, XREAL_1: 44;

        then t0 in [.(t0 - r), t0.];

        hence thesis by A8;

      end;

      

       A9: ((f3 (#) g1) - (f1 (#) g3)) is_left_differentiable_in t0 by A5, FDIFF_3: 22;

      

       A10: t0 in ( dom ((f3 (#) g1) - (f1 (#) g3)))

      proof

        consider r1 such that

         A11: 0 < r1 & [.(t0 - r1), t0.] c= ( dom ((f3 (#) g1) - (f1 (#) g3))) by A9, FDIFF_3:def 4;

        (t0 - r1) <= t0 by A11, XREAL_1: 44;

        then t0 in [.(t0 - r1), t0.];

        hence thesis by A11;

      end;

      

       A12: ((f1 (#) g2) - (f2 (#) g1)) is_left_differentiable_in t0 by A5, FDIFF_3: 22;

      

       A13: t0 in ( dom ((f1 (#) g2) - (f2 (#) g1)))

      proof

        consider r2 such that

         A14: 0 < r2 & [.(t0 - r2), t0.] c= ( dom ((f1 (#) g2) - (f2 (#) g1))) by A12, FDIFF_3:def 4;

        (t0 - r2) <= t0 by A14, XREAL_1: 44;

        then t0 in [.(t0 - r2), t0.];

        hence thesis by A14;

      end;

      ( VFuncdiff ((h1 (#) ((f2 (#) g3) - (f3 (#) g2))),(h2 (#) ((f3 (#) g1) - (f1 (#) g3))),(h3 (#) ((f1 (#) g2) - (f2 (#) g1))),t0)) = ( |[((((f2 (#) g3) - (f3 (#) g2)) . t0) * ( diff (h1,t0))), ((((f3 (#) g1) - (f1 (#) g3)) . t0) * ( diff (h2,t0))), ((((f1 (#) g2) - (f2 (#) g1)) . t0) * ( diff (h3,t0)))]| + |[((h1 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0))), ((h2 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0))), ((h3 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0)))]|) by A3, A5, Th91

      .= ( |[((((f2 (#) g3) - (f3 (#) g2)) . t0) * ( diff (h1,t0))), ((((f3 (#) g1) - (f1 (#) g3)) . t0) * ( diff (h2,t0))), ((((f1 (#) g2) - (f2 (#) g1)) . t0) * ( diff (h3,t0)))]| + |[((h1 . t0) * (( diff ((f2 (#) g3),t0)) - ( diff ((f3 (#) g2),t0)))), ((h2 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0))), ((h3 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0)))]|) by A4, FDIFF_1: 14

      .= ( |[((((f2 (#) g3) - (f3 (#) g2)) . t0) * ( diff (h1,t0))), ((((f3 (#) g1) - (f1 (#) g3)) . t0) * ( diff (h2,t0))), ((((f1 (#) g2) - (f2 (#) g1)) . t0) * ( diff (h3,t0)))]| + |[((h1 . t0) * (( diff ((f2 (#) g3),t0)) - ( diff ((f3 (#) g2),t0)))), ((h2 . t0) * (( diff ((f3 (#) g1),t0)) - ( diff ((f1 (#) g3),t0)))), ((h3 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0)))]|) by A4, FDIFF_1: 14

      .= ( |[((((f2 (#) g3) - (f3 (#) g2)) . t0) * ( diff (h1,t0))), ((((f3 (#) g1) - (f1 (#) g3)) . t0) * ( diff (h2,t0))), ((((f1 (#) g2) - (f2 (#) g1)) . t0) * ( diff (h3,t0)))]| + |[((h1 . t0) * (( diff ((f2 (#) g3),t0)) - ( diff ((f3 (#) g2),t0)))), ((h2 . t0) * (( diff ((f3 (#) g1),t0)) - ( diff ((f1 (#) g3),t0)))), ((h3 . t0) * (( diff ((f1 (#) g2),t0)) - ( diff ((f2 (#) g1),t0))))]|) by A4, FDIFF_1: 14

      .= ( |[((((f2 (#) g3) - (f3 (#) g2)) . t0) * ( diff (h1,t0))), ((((f3 (#) g1) - (f1 (#) g3)) . t0) * ( diff (h2,t0))), ((((f1 (#) g2) - (f2 (#) g1)) . t0) * ( diff (h3,t0)))]| + |[((h1 . t0) * ((((g3 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g3,t0)))) - ( diff ((f3 (#) g2),t0)))), ((h2 . t0) * (( diff ((f3 (#) g1),t0)) - ( diff ((f1 (#) g3),t0)))), ((h3 . t0) * (( diff ((f1 (#) g2),t0)) - ( diff ((f2 (#) g1),t0))))]|) by A1, A2, FDIFF_1: 16

      .= ( |[((((f2 (#) g3) - (f3 (#) g2)) . t0) * ( diff (h1,t0))), ((((f3 (#) g1) - (f1 (#) g3)) . t0) * ( diff (h2,t0))), ((((f1 (#) g2) - (f2 (#) g1)) . t0) * ( diff (h3,t0)))]| + |[((h1 . t0) * ((((g3 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g3,t0)))) - (((g2 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g2,t0)))))), ((h2 . t0) * (( diff ((f3 (#) g1),t0)) - ( diff ((f1 (#) g3),t0)))), ((h3 . t0) * (( diff ((f1 (#) g2),t0)) - ( diff ((f2 (#) g1),t0))))]|) by A1, A2, FDIFF_1: 16

      .= ( |[((((f2 (#) g3) - (f3 (#) g2)) . t0) * ( diff (h1,t0))), ((((f3 (#) g1) - (f1 (#) g3)) . t0) * ( diff (h2,t0))), ((((f1 (#) g2) - (f2 (#) g1)) . t0) * ( diff (h3,t0)))]| + |[((h1 . t0) * ((((g3 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g3,t0)))) - (((g2 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g2,t0)))))), ((h2 . t0) * ((((g1 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g1,t0)))) - ( diff ((f1 (#) g3),t0)))), ((h3 . t0) * (( diff ((f1 (#) g2),t0)) - ( diff ((f2 (#) g1),t0))))]|) by A1, A2, FDIFF_1: 16

      .= ( |[((((f2 (#) g3) - (f3 (#) g2)) . t0) * ( diff (h1,t0))), ((((f3 (#) g1) - (f1 (#) g3)) . t0) * ( diff (h2,t0))), ((((f1 (#) g2) - (f2 (#) g1)) . t0) * ( diff (h3,t0)))]| + |[((h1 . t0) * ((((g3 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g3,t0)))) - (((g2 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g2,t0)))))), ((h2 . t0) * ((((g1 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g1,t0)))) - (((g3 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g3,t0)))))), ((h3 . t0) * (( diff ((f1 (#) g2),t0)) - ( diff ((f2 (#) g1),t0))))]|) by A1, A2, FDIFF_1: 16

      .= ( |[((((f2 (#) g3) - (f3 (#) g2)) . t0) * ( diff (h1,t0))), ((((f3 (#) g1) - (f1 (#) g3)) . t0) * ( diff (h2,t0))), ((((f1 (#) g2) - (f2 (#) g1)) . t0) * ( diff (h3,t0)))]| + |[((h1 . t0) * ((((g3 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g3,t0)))) - (((g2 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g2,t0)))))), ((h2 . t0) * ((((g1 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g1,t0)))) - (((g3 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g3,t0)))))), ((h3 . t0) * ((((g2 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g2,t0)))) - ( diff ((f2 (#) g1),t0))))]|) by A1, A2, FDIFF_1: 16

      .= ( |[((((f2 (#) g3) - (f3 (#) g2)) . t0) * ( diff (h1,t0))), ((((f3 (#) g1) - (f1 (#) g3)) . t0) * ( diff (h2,t0))), ((((f1 (#) g2) - (f2 (#) g1)) . t0) * ( diff (h3,t0)))]| + |[((h1 . t0) * ((((g3 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g3,t0)))) - (((g2 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g2,t0)))))), ((h2 . t0) * ((((g1 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g1,t0)))) - (((g3 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g3,t0)))))), ((h3 . t0) * ((((g2 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g2,t0)))) - (((g1 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g1,t0))))))]|) by A1, A2, FDIFF_1: 16

      .= ( |[((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * ( diff (h1,t0))), ((((f3 (#) g1) - (f1 (#) g3)) . t0) * ( diff (h2,t0))), ((((f1 (#) g2) - (f2 (#) g1)) . t0) * ( diff (h3,t0)))]| + |[((h1 . t0) * ((((g3 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g3,t0)))) - (((g2 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g2,t0)))))), ((h2 . t0) * ((((g1 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g1,t0)))) - (((g3 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g3,t0)))))), ((h3 . t0) * ((((g2 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g2,t0)))) - (((g1 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g1,t0))))))]|) by A7, VALUED_1: 13

      .= ( |[((((f2 . t0) * (g3 . t0)) - ((f3 (#) g2) . t0)) * ( diff (h1,t0))), ((((f3 (#) g1) - (f1 (#) g3)) . t0) * ( diff (h2,t0))), ((((f1 (#) g2) - (f2 (#) g1)) . t0) * ( diff (h3,t0)))]| + |[((h1 . t0) * ((((g3 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g3,t0)))) - (((g2 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g2,t0)))))), ((h2 . t0) * ((((g1 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g1,t0)))) - (((g3 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g3,t0)))))), ((h3 . t0) * ((((g2 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g2,t0)))) - (((g1 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g1,t0))))))]|) by VALUED_1: 5

      .= ( |[((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h1,t0))), ((((f3 (#) g1) - (f1 (#) g3)) . t0) * ( diff (h2,t0))), ((((f1 (#) g2) - (f2 (#) g1)) . t0) * ( diff (h3,t0)))]| + |[((h1 . t0) * ((((g3 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g3,t0)))) - (((g2 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g2,t0)))))), ((h2 . t0) * ((((g1 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g1,t0)))) - (((g3 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g3,t0)))))), ((h3 . t0) * ((((g2 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g2,t0)))) - (((g1 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g1,t0))))))]|) by VALUED_1: 5

      .= ( |[((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h1,t0))), ((((f3 (#) g1) . t0) - ((f1 (#) g3) . t0)) * ( diff (h2,t0))), ((((f1 (#) g2) - (f2 (#) g1)) . t0) * ( diff (h3,t0)))]| + |[((h1 . t0) * ((((g3 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g3,t0)))) - (((g2 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g2,t0)))))), ((h2 . t0) * ((((g1 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g1,t0)))) - (((g3 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g3,t0)))))), ((h3 . t0) * ((((g2 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g2,t0)))) - (((g1 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g1,t0))))))]|) by A10, VALUED_1: 13

      .= ( |[((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h1,t0))), ((((f3 . t0) * (g1 . t0)) - ((f1 (#) g3) . t0)) * ( diff (h2,t0))), ((((f1 (#) g2) - (f2 (#) g1)) . t0) * ( diff (h3,t0)))]| + |[((h1 . t0) * ((((g3 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g3,t0)))) - (((g2 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g2,t0)))))), ((h2 . t0) * ((((g1 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g1,t0)))) - (((g3 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g3,t0)))))), ((h3 . t0) * ((((g2 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g2,t0)))) - (((g1 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g1,t0))))))]|) by VALUED_1: 5

      .= ( |[((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h1,t0))), ((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h2,t0))), ((((f1 (#) g2) - (f2 (#) g1)) . t0) * ( diff (h3,t0)))]| + |[((h1 . t0) * ((((g3 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g3,t0)))) - (((g2 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g2,t0)))))), ((h2 . t0) * ((((g1 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g1,t0)))) - (((g3 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g3,t0)))))), ((h3 . t0) * ((((g2 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g2,t0)))) - (((g1 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g1,t0))))))]|) by VALUED_1: 5

      .= ( |[((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h1,t0))), ((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h2,t0))), ((((f1 (#) g2) . t0) - ((f2 (#) g1) . t0)) * ( diff (h3,t0)))]| + |[((h1 . t0) * ((((g3 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g3,t0)))) - (((g2 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g2,t0)))))), ((h2 . t0) * ((((g1 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g1,t0)))) - (((g3 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g3,t0)))))), ((h3 . t0) * ((((g2 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g2,t0)))) - (((g1 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g1,t0))))))]|) by A13, VALUED_1: 13

      .= ( |[((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h1,t0))), ((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h2,t0))), ((((f1 . t0) * (g2 . t0)) - ((f2 (#) g1) . t0)) * ( diff (h3,t0)))]| + |[((h1 . t0) * ((((g3 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g3,t0)))) - (((g2 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g2,t0)))))), ((h2 . t0) * ((((g1 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g1,t0)))) - (((g3 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g3,t0)))))), ((h3 . t0) * ((((g2 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g2,t0)))) - (((g1 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g1,t0))))))]|) by VALUED_1: 5

      .= ( |[(( diff (h1,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))), (( diff (h2,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0)))), (( diff (h3,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))))]| + |[(((((h1 . t0) * (g3 . t0)) * ( diff (f2,t0))) - (((h1 . t0) * (g2 . t0)) * ( diff (f3,t0)))) + ((((h1 . t0) * (f2 . t0)) * ( diff (g3,t0))) - (((h1 . t0) * (f3 . t0)) * ( diff (g2,t0))))), (((((h2 . t0) * (g1 . t0)) * ( diff (f3,t0))) - (((h2 . t0) * (g3 . t0)) * ( diff (f1,t0)))) + ((((h2 . t0) * (f3 . t0)) * ( diff (g1,t0))) - (((h2 . t0) * (f1 . t0)) * ( diff (g3,t0))))), (((((h3 . t0) * (g2 . t0)) * ( diff (f1,t0))) - (((h3 . t0) * (g1 . t0)) * ( diff (f2,t0)))) + ((((h3 . t0) * (f1 . t0)) * ( diff (g2,t0))) - (((h3 . t0) * (f2 . t0)) * ( diff (g1,t0)))))]|) by VALUED_1: 5

      .= ( |[(( diff (h1,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))), (( diff (h2,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0)))), (( diff (h3,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))))]| + ( |[((((h1 . t0) * (g3 . t0)) * ( diff (f2,t0))) - (((h1 . t0) * (g2 . t0)) * ( diff (f3,t0)))), ((((h2 . t0) * (g1 . t0)) * ( diff (f3,t0))) - (((h2 . t0) * (g3 . t0)) * ( diff (f1,t0)))), ((((h3 . t0) * (g2 . t0)) * ( diff (f1,t0))) - (((h3 . t0) * (g1 . t0)) * ( diff (f2,t0))))]| + |[((((h1 . t0) * (f2 . t0)) * ( diff (g3,t0))) - (((h1 . t0) * (f3 . t0)) * ( diff (g2,t0)))), ((((h2 . t0) * (f3 . t0)) * ( diff (g1,t0))) - (((h2 . t0) * (f1 . t0)) * ( diff (g3,t0)))), ((((h3 . t0) * (f1 . t0)) * ( diff (g2,t0))) - (((h3 . t0) * (f2 . t0)) * ( diff (g1,t0))))]|)) by Lm8

      .= (( |[(( diff (h1,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))), (( diff (h2,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0)))), (( diff (h3,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))))]| + |[((h1 . t0) * ((( diff (f2,t0)) * (g3 . t0)) - (( diff (f3,t0)) * (g2 . t0)))), ((h2 . t0) * ((( diff (f3,t0)) * (g1 . t0)) - (( diff (f1,t0)) * (g3 . t0)))), ((h3 . t0) * ((( diff (f1,t0)) * (g2 . t0)) - (( diff (f2,t0)) * (g1 . t0))))]|) + |[((h1 . t0) * (((f2 . t0) * ( diff (g3,t0))) - ((f3 . t0) * ( diff (g2,t0))))), ((h2 . t0) * (((f3 . t0) * ( diff (g1,t0))) - ((f1 . t0) * ( diff (g3,t0))))), ((h3 . t0) * (((f1 . t0) * ( diff (g2,t0))) - ((f2 . t0) * ( diff (g1,t0)))))]|) by RVSUM_1: 15;

      hence thesis;

    end;

    theorem :: EUCLID_8:113

    (f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0) & (g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0) & (h1 is_differentiable_in t0 & h2 is_differentiable_in t0 & h3 is_differentiable_in t0) implies ( VFuncdiff ((((h2 (#) f2) (#) g3) - ((h3 (#) f3) (#) g2)),(((h3 (#) f3) (#) g1) - ((h1 (#) f1) (#) g3)),(((h1 (#) f1) (#) g2) - ((h2 (#) f2) (#) g1)),t0)) = (( |[((((h2 . t0) * (f2 . t0)) * ( diff (g3,t0))) - (((h3 . t0) * (f3 . t0)) * ( diff (g2,t0)))), ((((h3 . t0) * (f3 . t0)) * ( diff (g1,t0))) - (((h1 . t0) * (f1 . t0)) * ( diff (g3,t0)))), ((((h1 . t0) * (f1 . t0)) * ( diff (g2,t0))) - (((h2 . t0) * (f2 . t0)) * ( diff (g1,t0))))]| + |[((((h2 . t0) * ( diff (f2,t0))) * (g3 . t0)) - (((h3 . t0) * ( diff (f3,t0))) * (g2 . t0))), ((((h3 . t0) * ( diff (f3,t0))) * (g1 . t0)) - (((h1 . t0) * ( diff (f1,t0))) * (g3 . t0))), ((((h1 . t0) * ( diff (f1,t0))) * (g2 . t0)) - (((h2 . t0) * ( diff (f2,t0))) * (g1 . t0)))]|) + |[(((( diff (h2,t0)) * (f2 . t0)) * (g3 . t0)) - ((( diff (h3,t0)) * (f3 . t0)) * (g2 . t0))), (((( diff (h3,t0)) * (f3 . t0)) * (g1 . t0)) - ((( diff (h1,t0)) * (f1 . t0)) * (g3 . t0))), (((( diff (h1,t0)) * (f1 . t0)) * (g2 . t0)) - ((( diff (h2,t0)) * (f2 . t0)) * (g1 . t0)))]|)

    proof

      assume that

       A1: f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 and

       A2: g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 and

       A3: h1 is_differentiable_in t0 & h2 is_differentiable_in t0 & h3 is_differentiable_in t0;

      

       A4: (h3 (#) f3) is_differentiable_in t0 & (h1 (#) f1) is_differentiable_in t0 & (h2 (#) f2) is_differentiable_in t0 by A1, A3, FDIFF_1: 16;

      then

       A5: ((h3 (#) f3) (#) g1) is_differentiable_in t0 & ((h3 (#) f3) (#) g2) is_differentiable_in t0 & ((h1 (#) f1) (#) g2) is_differentiable_in t0 & ((h1 (#) f1) (#) g3) is_differentiable_in t0 & ((h2 (#) f2) (#) g3) is_differentiable_in t0 & ((h2 (#) f2) (#) g1) is_differentiable_in t0 by A2, FDIFF_1: 16;

      ( VFuncdiff ((((h2 (#) f2) (#) g3) - ((h3 (#) f3) (#) g2)),(((h3 (#) f3) (#) g1) - ((h1 (#) f1) (#) g3)),(((h1 (#) f1) (#) g2) - ((h2 (#) f2) (#) g1)),t0)) = (( VFuncdiff (((h2 (#) f2) (#) g3),((h3 (#) f3) (#) g1),((h1 (#) f1) (#) g2),t0)) - ( VFuncdiff (((h3 (#) f3) (#) g2),((h1 (#) f1) (#) g3),((h2 (#) f2) (#) g1),t0))) by A5, Th89

      .= (( |[((g3 . t0) * ( diff ((h2 (#) f2),t0))), ((g1 . t0) * ( diff ((h3 (#) f3),t0))), ((g2 . t0) * ( diff ((h1 (#) f1),t0)))]| + |[(((h2 (#) f2) . t0) * ( diff (g3,t0))), (((h3 (#) f3) . t0) * ( diff (g1,t0))), (((h1 (#) f1) . t0) * ( diff (g2,t0)))]|) - ( VFuncdiff (((h3 (#) f3) (#) g2),((h1 (#) f1) (#) g3),((h2 (#) f2) (#) g1),t0))) by A2, A4, Th91

      .= (( |[((g3 . t0) * ( diff ((h2 (#) f2),t0))), ((g1 . t0) * ( diff ((h3 (#) f3),t0))), ((g2 . t0) * ( diff ((h1 (#) f1),t0)))]| + |[(((h2 (#) f2) . t0) * ( diff (g3,t0))), (((h3 (#) f3) . t0) * ( diff (g1,t0))), (((h1 (#) f1) . t0) * ( diff (g2,t0)))]|) - ( |[((g2 . t0) * ( diff ((h3 (#) f3),t0))), ((g3 . t0) * ( diff ((h1 (#) f1),t0))), ((g1 . t0) * ( diff ((h2 (#) f2),t0)))]| + |[(((h3 (#) f3) . t0) * ( diff (g2,t0))), (((h1 (#) f1) . t0) * ( diff (g3,t0))), (((h2 (#) f2) . t0) * ( diff (g1,t0)))]|)) by A2, A4, Th91

      .= (( |[((g3 . t0) * (((f2 . t0) * ( diff (h2,t0))) + ((h2 . t0) * ( diff (f2,t0))))), ((g1 . t0) * ( diff ((h3 (#) f3),t0))), ((g2 . t0) * ( diff ((h1 (#) f1),t0)))]| + |[(((h2 (#) f2) . t0) * ( diff (g3,t0))), (((h3 (#) f3) . t0) * ( diff (g1,t0))), (((h1 (#) f1) . t0) * ( diff (g2,t0)))]|) - ( |[((g2 . t0) * ( diff ((h3 (#) f3),t0))), ((g3 . t0) * ( diff ((h1 (#) f1),t0))), ((g1 . t0) * ( diff ((h2 (#) f2),t0)))]| + |[(((h3 (#) f3) . t0) * ( diff (g2,t0))), (((h1 (#) f1) . t0) * ( diff (g3,t0))), (((h2 (#) f2) . t0) * ( diff (g1,t0)))]|)) by A1, A3, FDIFF_1: 16

      .= (( |[((g3 . t0) * (((f2 . t0) * ( diff (h2,t0))) + ((h2 . t0) * ( diff (f2,t0))))), ((g1 . t0) * (((f3 . t0) * ( diff (h3,t0))) + ((h3 . t0) * ( diff (f3,t0))))), ((g2 . t0) * ( diff ((h1 (#) f1),t0)))]| + |[(((h2 (#) f2) . t0) * ( diff (g3,t0))), (((h3 (#) f3) . t0) * ( diff (g1,t0))), (((h1 (#) f1) . t0) * ( diff (g2,t0)))]|) - ( |[((g2 . t0) * ( diff ((h3 (#) f3),t0))), ((g3 . t0) * ( diff ((h1 (#) f1),t0))), ((g1 . t0) * ( diff ((h2 (#) f2),t0)))]| + |[(((h3 (#) f3) . t0) * ( diff (g2,t0))), (((h1 (#) f1) . t0) * ( diff (g3,t0))), (((h2 (#) f2) . t0) * ( diff (g1,t0)))]|)) by A1, A3, FDIFF_1: 16

      .= (( |[((g3 . t0) * (((f2 . t0) * ( diff (h2,t0))) + ((h2 . t0) * ( diff (f2,t0))))), ((g1 . t0) * (((f3 . t0) * ( diff (h3,t0))) + ((h3 . t0) * ( diff (f3,t0))))), ((g2 . t0) * (((f1 . t0) * ( diff (h1,t0))) + ((h1 . t0) * ( diff (f1,t0)))))]| + |[(((h2 (#) f2) . t0) * ( diff (g3,t0))), (((h3 (#) f3) . t0) * ( diff (g1,t0))), (((h1 (#) f1) . t0) * ( diff (g2,t0)))]|) - ( |[((g2 . t0) * ( diff ((h3 (#) f3),t0))), ((g3 . t0) * ( diff ((h1 (#) f1),t0))), ((g1 . t0) * ( diff ((h2 (#) f2),t0)))]| + |[(((h3 (#) f3) . t0) * ( diff (g2,t0))), (((h1 (#) f1) . t0) * ( diff (g3,t0))), (((h2 (#) f2) . t0) * ( diff (g1,t0)))]|)) by A1, A3, FDIFF_1: 16

      .= (( |[((g3 . t0) * (((f2 . t0) * ( diff (h2,t0))) + ((h2 . t0) * ( diff (f2,t0))))), ((g1 . t0) * (((f3 . t0) * ( diff (h3,t0))) + ((h3 . t0) * ( diff (f3,t0))))), ((g2 . t0) * (((f1 . t0) * ( diff (h1,t0))) + ((h1 . t0) * ( diff (f1,t0)))))]| + |[(((h2 . t0) * (f2 . t0)) * ( diff (g3,t0))), (((h3 (#) f3) . t0) * ( diff (g1,t0))), (((h1 (#) f1) . t0) * ( diff (g2,t0)))]|) - ( |[((g2 . t0) * ( diff ((h3 (#) f3),t0))), ((g3 . t0) * ( diff ((h1 (#) f1),t0))), ((g1 . t0) * ( diff ((h2 (#) f2),t0)))]| + |[(((h3 (#) f3) . t0) * ( diff (g2,t0))), (((h1 (#) f1) . t0) * ( diff (g3,t0))), (((h2 (#) f2) . t0) * ( diff (g1,t0)))]|)) by VALUED_1: 5

      .= (( |[((g3 . t0) * (((f2 . t0) * ( diff (h2,t0))) + ((h2 . t0) * ( diff (f2,t0))))), ((g1 . t0) * (((f3 . t0) * ( diff (h3,t0))) + ((h3 . t0) * ( diff (f3,t0))))), ((g2 . t0) * (((f1 . t0) * ( diff (h1,t0))) + ((h1 . t0) * ( diff (f1,t0)))))]| + |[(((h2 . t0) * (f2 . t0)) * ( diff (g3,t0))), (((h3 . t0) * (f3 . t0)) * ( diff (g1,t0))), (((h1 (#) f1) . t0) * ( diff (g2,t0)))]|) - ( |[((g2 . t0) * ( diff ((h3 (#) f3),t0))), ((g3 . t0) * ( diff ((h1 (#) f1),t0))), ((g1 . t0) * ( diff ((h2 (#) f2),t0)))]| + |[(((h3 (#) f3) . t0) * ( diff (g2,t0))), (((h1 (#) f1) . t0) * ( diff (g3,t0))), (((h2 (#) f2) . t0) * ( diff (g1,t0)))]|)) by VALUED_1: 5

      .= (( |[((g3 . t0) * (((f2 . t0) * ( diff (h2,t0))) + ((h2 . t0) * ( diff (f2,t0))))), ((g1 . t0) * (((f3 . t0) * ( diff (h3,t0))) + ((h3 . t0) * ( diff (f3,t0))))), ((g2 . t0) * (((f1 . t0) * ( diff (h1,t0))) + ((h1 . t0) * ( diff (f1,t0)))))]| + |[(((h2 . t0) * (f2 . t0)) * ( diff (g3,t0))), (((h3 . t0) * (f3 . t0)) * ( diff (g1,t0))), (((h1 . t0) * (f1 . t0)) * ( diff (g2,t0)))]|) - ( |[((g2 . t0) * ( diff ((h3 (#) f3),t0))), ((g3 . t0) * ( diff ((h1 (#) f1),t0))), ((g1 . t0) * ( diff ((h2 (#) f2),t0)))]| + |[(((h3 (#) f3) . t0) * ( diff (g2,t0))), (((h1 (#) f1) . t0) * ( diff (g3,t0))), (((h2 (#) f2) . t0) * ( diff (g1,t0)))]|)) by VALUED_1: 5

      .= (( |[((g3 . t0) * (((f2 . t0) * ( diff (h2,t0))) + ((h2 . t0) * ( diff (f2,t0))))), ((g1 . t0) * (((f3 . t0) * ( diff (h3,t0))) + ((h3 . t0) * ( diff (f3,t0))))), ((g2 . t0) * (((f1 . t0) * ( diff (h1,t0))) + ((h1 . t0) * ( diff (f1,t0)))))]| + |[(((h2 . t0) * (f2 . t0)) * ( diff (g3,t0))), (((h3 . t0) * (f3 . t0)) * ( diff (g1,t0))), (((h1 . t0) * (f1 . t0)) * ( diff (g2,t0)))]|) - ( |[((g2 . t0) * ( diff ((h3 (#) f3),t0))), ((g3 . t0) * ( diff ((h1 (#) f1),t0))), ((g1 . t0) * ( diff ((h2 (#) f2),t0)))]| + |[(((h3 . t0) * (f3 . t0)) * ( diff (g2,t0))), (((h1 (#) f1) . t0) * ( diff (g3,t0))), (((h2 (#) f2) . t0) * ( diff (g1,t0)))]|)) by VALUED_1: 5

      .= (( |[((g3 . t0) * (((f2 . t0) * ( diff (h2,t0))) + ((h2 . t0) * ( diff (f2,t0))))), ((g1 . t0) * (((f3 . t0) * ( diff (h3,t0))) + ((h3 . t0) * ( diff (f3,t0))))), ((g2 . t0) * (((f1 . t0) * ( diff (h1,t0))) + ((h1 . t0) * ( diff (f1,t0)))))]| + |[(((h2 . t0) * (f2 . t0)) * ( diff (g3,t0))), (((h3 . t0) * (f3 . t0)) * ( diff (g1,t0))), (((h1 . t0) * (f1 . t0)) * ( diff (g2,t0)))]|) - ( |[((g2 . t0) * ( diff ((h3 (#) f3),t0))), ((g3 . t0) * ( diff ((h1 (#) f1),t0))), ((g1 . t0) * ( diff ((h2 (#) f2),t0)))]| + |[(((h3 . t0) * (f3 . t0)) * ( diff (g2,t0))), (((h1 . t0) * (f1 . t0)) * ( diff (g3,t0))), (((h2 (#) f2) . t0) * ( diff (g1,t0)))]|)) by VALUED_1: 5

      .= (( |[((g3 . t0) * (((f2 . t0) * ( diff (h2,t0))) + ((h2 . t0) * ( diff (f2,t0))))), ((g1 . t0) * (((f3 . t0) * ( diff (h3,t0))) + ((h3 . t0) * ( diff (f3,t0))))), ((g2 . t0) * (((f1 . t0) * ( diff (h1,t0))) + ((h1 . t0) * ( diff (f1,t0)))))]| + |[(((h2 . t0) * (f2 . t0)) * ( diff (g3,t0))), (((h3 . t0) * (f3 . t0)) * ( diff (g1,t0))), (((h1 . t0) * (f1 . t0)) * ( diff (g2,t0)))]|) - ( |[((g2 . t0) * ( diff ((h3 (#) f3),t0))), ((g3 . t0) * ( diff ((h1 (#) f1),t0))), ((g1 . t0) * ( diff ((h2 (#) f2),t0)))]| + |[(((h3 . t0) * (f3 . t0)) * ( diff (g2,t0))), (((h1 . t0) * (f1 . t0)) * ( diff (g3,t0))), (((h2 . t0) * (f2 . t0)) * ( diff (g1,t0)))]|)) by VALUED_1: 5

      .= (( |[((g3 . t0) * (((f2 . t0) * ( diff (h2,t0))) + ((h2 . t0) * ( diff (f2,t0))))), ((g1 . t0) * (((f3 . t0) * ( diff (h3,t0))) + ((h3 . t0) * ( diff (f3,t0))))), ((g2 . t0) * (((f1 . t0) * ( diff (h1,t0))) + ((h1 . t0) * ( diff (f1,t0)))))]| + |[(((h2 . t0) * (f2 . t0)) * ( diff (g3,t0))), (((h3 . t0) * (f3 . t0)) * ( diff (g1,t0))), (((h1 . t0) * (f1 . t0)) * ( diff (g2,t0)))]|) - ( |[((g2 . t0) * (((f3 . t0) * ( diff (h3,t0))) + ((h3 . t0) * ( diff (f3,t0))))), ((g3 . t0) * ( diff ((h1 (#) f1),t0))), ((g1 . t0) * ( diff ((h2 (#) f2),t0)))]| + |[(((h3 . t0) * (f3 . t0)) * ( diff (g2,t0))), (((h1 . t0) * (f1 . t0)) * ( diff (g3,t0))), (((h2 . t0) * (f2 . t0)) * ( diff (g1,t0)))]|)) by A1, A3, FDIFF_1: 16

      .= (( |[((g3 . t0) * (((f2 . t0) * ( diff (h2,t0))) + ((h2 . t0) * ( diff (f2,t0))))), ((g1 . t0) * (((f3 . t0) * ( diff (h3,t0))) + ((h3 . t0) * ( diff (f3,t0))))), ((g2 . t0) * (((f1 . t0) * ( diff (h1,t0))) + ((h1 . t0) * ( diff (f1,t0)))))]| + |[(((h2 . t0) * (f2 . t0)) * ( diff (g3,t0))), (((h3 . t0) * (f3 . t0)) * ( diff (g1,t0))), (((h1 . t0) * (f1 . t0)) * ( diff (g2,t0)))]|) - ( |[((g2 . t0) * (((f3 . t0) * ( diff (h3,t0))) + ((h3 . t0) * ( diff (f3,t0))))), ((g3 . t0) * (((f1 . t0) * ( diff (h1,t0))) + ((h1 . t0) * ( diff (f1,t0))))), ((g1 . t0) * ( diff ((h2 (#) f2),t0)))]| + |[(((h3 . t0) * (f3 . t0)) * ( diff (g2,t0))), (((h1 . t0) * (f1 . t0)) * ( diff (g3,t0))), (((h2 . t0) * (f2 . t0)) * ( diff (g1,t0)))]|)) by A1, A3, FDIFF_1: 16

      .= (( |[((g3 . t0) * (((f2 . t0) * ( diff (h2,t0))) + ((h2 . t0) * ( diff (f2,t0))))), ((g1 . t0) * (((f3 . t0) * ( diff (h3,t0))) + ((h3 . t0) * ( diff (f3,t0))))), ((g2 . t0) * (((f1 . t0) * ( diff (h1,t0))) + ((h1 . t0) * ( diff (f1,t0)))))]| + |[(((h2 . t0) * (f2 . t0)) * ( diff (g3,t0))), (((h3 . t0) * (f3 . t0)) * ( diff (g1,t0))), (((h1 . t0) * (f1 . t0)) * ( diff (g2,t0)))]|) - ( |[((g2 . t0) * (((f3 . t0) * ( diff (h3,t0))) + ((h3 . t0) * ( diff (f3,t0))))), ((g3 . t0) * (((f1 . t0) * ( diff (h1,t0))) + ((h1 . t0) * ( diff (f1,t0))))), ((g1 . t0) * (((f2 . t0) * ( diff (h2,t0))) + ((h2 . t0) * ( diff (f2,t0)))))]| + |[(((h3 . t0) * (f3 . t0)) * ( diff (g2,t0))), (((h1 . t0) * (f1 . t0)) * ( diff (g3,t0))), (((h2 . t0) * (f2 . t0)) * ( diff (g1,t0)))]|)) by A1, A3, FDIFF_1: 16

      .= ((( |[((g3 . t0) * (((f2 . t0) * ( diff (h2,t0))) + ((h2 . t0) * ( diff (f2,t0))))), ((g1 . t0) * (((f3 . t0) * ( diff (h3,t0))) + ((h3 . t0) * ( diff (f3,t0))))), ((g2 . t0) * (((f1 . t0) * ( diff (h1,t0))) + ((h1 . t0) * ( diff (f1,t0)))))]| + |[(((h2 . t0) * (f2 . t0)) * ( diff (g3,t0))), (((h3 . t0) * (f3 . t0)) * ( diff (g1,t0))), (((h1 . t0) * (f1 . t0)) * ( diff (g2,t0)))]|) - |[((g2 . t0) * (((f3 . t0) * ( diff (h3,t0))) + ((h3 . t0) * ( diff (f3,t0))))), ((g3 . t0) * (((f1 . t0) * ( diff (h1,t0))) + ((h1 . t0) * ( diff (f1,t0))))), ((g1 . t0) * (((f2 . t0) * ( diff (h2,t0))) + ((h2 . t0) * ( diff (f2,t0)))))]|) - |[(((h3 . t0) * (f3 . t0)) * ( diff (g2,t0))), (((h1 . t0) * (f1 . t0)) * ( diff (g3,t0))), (((h2 . t0) * (f2 . t0)) * ( diff (g1,t0)))]|) by RVSUM_1: 39

      .= ((( |[((g3 . t0) * (((f2 . t0) * ( diff (h2,t0))) + ((h2 . t0) * ( diff (f2,t0))))), ((g1 . t0) * (((f3 . t0) * ( diff (h3,t0))) + ((h3 . t0) * ( diff (f3,t0))))), ((g2 . t0) * (((f1 . t0) * ( diff (h1,t0))) + ((h1 . t0) * ( diff (f1,t0)))))]| - |[((g2 . t0) * (((f3 . t0) * ( diff (h3,t0))) + ((h3 . t0) * ( diff (f3,t0))))), ((g3 . t0) * (((f1 . t0) * ( diff (h1,t0))) + ((h1 . t0) * ( diff (f1,t0))))), ((g1 . t0) * (((f2 . t0) * ( diff (h2,t0))) + ((h2 . t0) * ( diff (f2,t0)))))]|) + |[(((h2 . t0) * (f2 . t0)) * ( diff (g3,t0))), (((h3 . t0) * (f3 . t0)) * ( diff (g1,t0))), (((h1 . t0) * (f1 . t0)) * ( diff (g2,t0)))]|) + ( - |[(((h3 . t0) * (f3 . t0)) * ( diff (g2,t0))), (((h1 . t0) * (f1 . t0)) * ( diff (g3,t0))), (((h2 . t0) * (f2 . t0)) * ( diff (g1,t0)))]|)) by RVSUM_1: 15

      .= (( |[((g3 . t0) * (((f2 . t0) * ( diff (h2,t0))) + ((h2 . t0) * ( diff (f2,t0))))), ((g1 . t0) * (((f3 . t0) * ( diff (h3,t0))) + ((h3 . t0) * ( diff (f3,t0))))), ((g2 . t0) * (((f1 . t0) * ( diff (h1,t0))) + ((h1 . t0) * ( diff (f1,t0)))))]| - |[((g2 . t0) * (((f3 . t0) * ( diff (h3,t0))) + ((h3 . t0) * ( diff (f3,t0))))), ((g3 . t0) * (((f1 . t0) * ( diff (h1,t0))) + ((h1 . t0) * ( diff (f1,t0))))), ((g1 . t0) * (((f2 . t0) * ( diff (h2,t0))) + ((h2 . t0) * ( diff (f2,t0)))))]|) + ( |[(((h2 . t0) * (f2 . t0)) * ( diff (g3,t0))), (((h3 . t0) * (f3 . t0)) * ( diff (g1,t0))), (((h1 . t0) * (f1 . t0)) * ( diff (g2,t0)))]| - |[(((h3 . t0) * (f3 . t0)) * ( diff (g2,t0))), (((h1 . t0) * (f1 . t0)) * ( diff (g3,t0))), (((h2 . t0) * (f2 . t0)) * ( diff (g1,t0)))]|)) by RVSUM_1: 15

      .= (( |[((g3 . t0) * (((f2 . t0) * ( diff (h2,t0))) + ((h2 . t0) * ( diff (f2,t0))))), ((g1 . t0) * (((f3 . t0) * ( diff (h3,t0))) + ((h3 . t0) * ( diff (f3,t0))))), ((g2 . t0) * (((f1 . t0) * ( diff (h1,t0))) + ((h1 . t0) * ( diff (f1,t0)))))]| - |[((g2 . t0) * (((f3 . t0) * ( diff (h3,t0))) + ((h3 . t0) * ( diff (f3,t0))))), ((g3 . t0) * (((f1 . t0) * ( diff (h1,t0))) + ((h1 . t0) * ( diff (f1,t0))))), ((g1 . t0) * (((f2 . t0) * ( diff (h2,t0))) + ((h2 . t0) * ( diff (f2,t0)))))]|) + |[((((h2 . t0) * (f2 . t0)) * ( diff (g3,t0))) - (((h3 . t0) * (f3 . t0)) * ( diff (g2,t0)))), ((((h3 . t0) * (f3 . t0)) * ( diff (g1,t0))) - (((h1 . t0) * (f1 . t0)) * ( diff (g3,t0)))), ((((h1 . t0) * (f1 . t0)) * ( diff (g2,t0))) - (((h2 . t0) * (f2 . t0)) * ( diff (g1,t0))))]|) by Lm11

      .= ( |[((((h2 . t0) * (f2 . t0)) * ( diff (g3,t0))) - (((h3 . t0) * (f3 . t0)) * ( diff (g2,t0)))), ((((h3 . t0) * (f3 . t0)) * ( diff (g1,t0))) - (((h1 . t0) * (f1 . t0)) * ( diff (g3,t0)))), ((((h1 . t0) * (f1 . t0)) * ( diff (g2,t0))) - (((h2 . t0) * (f2 . t0)) * ( diff (g1,t0))))]| + |[(((g3 . t0) * (((f2 . t0) * ( diff (h2,t0))) + ((h2 . t0) * ( diff (f2,t0))))) - ((g2 . t0) * (((f3 . t0) * ( diff (h3,t0))) + ((h3 . t0) * ( diff (f3,t0)))))), (((g1 . t0) * (((f3 . t0) * ( diff (h3,t0))) + ((h3 . t0) * ( diff (f3,t0))))) - ((g3 . t0) * (((f1 . t0) * ( diff (h1,t0))) + ((h1 . t0) * ( diff (f1,t0)))))), (((g2 . t0) * (((f1 . t0) * ( diff (h1,t0))) + ((h1 . t0) * ( diff (f1,t0))))) - ((g1 . t0) * (((f2 . t0) * ( diff (h2,t0))) + ((h2 . t0) * ( diff (f2,t0))))))]|) by Lm11

      .= ( |[((((h2 . t0) * (f2 . t0)) * ( diff (g3,t0))) - (((h3 . t0) * (f3 . t0)) * ( diff (g2,t0)))), ((((h3 . t0) * (f3 . t0)) * ( diff (g1,t0))) - (((h1 . t0) * (f1 . t0)) * ( diff (g3,t0)))), ((((h1 . t0) * (f1 . t0)) * ( diff (g2,t0))) - (((h2 . t0) * (f2 . t0)) * ( diff (g1,t0))))]| + |[(((((g3 . t0) * (h2 . t0)) * ( diff (f2,t0))) - (((g2 . t0) * (h3 . t0)) * ( diff (f3,t0)))) + ((((g3 . t0) * (f2 . t0)) * ( diff (h2,t0))) - (((g2 . t0) * (f3 . t0)) * ( diff (h3,t0))))), (((((g1 . t0) * (h3 . t0)) * ( diff (f3,t0))) - (((g3 . t0) * (h1 . t0)) * ( diff (f1,t0)))) + ((((g1 . t0) * (f3 . t0)) * ( diff (h3,t0))) - (((g3 . t0) * (f1 . t0)) * ( diff (h1,t0))))), (((((g2 . t0) * (h1 . t0)) * ( diff (f1,t0))) - (((g1 . t0) * (h2 . t0)) * ( diff (f2,t0)))) + ((((g2 . t0) * (f1 . t0)) * ( diff (h1,t0))) - (((g1 . t0) * (f2 . t0)) * ( diff (h2,t0)))))]|)

      .= ( |[((((h2 . t0) * (f2 . t0)) * ( diff (g3,t0))) - (((h3 . t0) * (f3 . t0)) * ( diff (g2,t0)))), ((((h3 . t0) * (f3 . t0)) * ( diff (g1,t0))) - (((h1 . t0) * (f1 . t0)) * ( diff (g3,t0)))), ((((h1 . t0) * (f1 . t0)) * ( diff (g2,t0))) - (((h2 . t0) * (f2 . t0)) * ( diff (g1,t0))))]| + ( |[((((g3 . t0) * (h2 . t0)) * ( diff (f2,t0))) - (((g2 . t0) * (h3 . t0)) * ( diff (f3,t0)))), ((((g1 . t0) * (h3 . t0)) * ( diff (f3,t0))) - (((g3 . t0) * (h1 . t0)) * ( diff (f1,t0)))), ((((g2 . t0) * (h1 . t0)) * ( diff (f1,t0))) - (((g1 . t0) * (h2 . t0)) * ( diff (f2,t0))))]| + |[((((g3 . t0) * (f2 . t0)) * ( diff (h2,t0))) - (((g2 . t0) * (f3 . t0)) * ( diff (h3,t0)))), ((((g1 . t0) * (f3 . t0)) * ( diff (h3,t0))) - (((g3 . t0) * (f1 . t0)) * ( diff (h1,t0)))), ((((g2 . t0) * (f1 . t0)) * ( diff (h1,t0))) - (((g1 . t0) * (f2 . t0)) * ( diff (h2,t0))))]|)) by Lm8

      .= (( |[((((h2 . t0) * (f2 . t0)) * ( diff (g3,t0))) - (((h3 . t0) * (f3 . t0)) * ( diff (g2,t0)))), ((((h3 . t0) * (f3 . t0)) * ( diff (g1,t0))) - (((h1 . t0) * (f1 . t0)) * ( diff (g3,t0)))), ((((h1 . t0) * (f1 . t0)) * ( diff (g2,t0))) - (((h2 . t0) * (f2 . t0)) * ( diff (g1,t0))))]| + |[((((g3 . t0) * (h2 . t0)) * ( diff (f2,t0))) - (((g2 . t0) * (h3 . t0)) * ( diff (f3,t0)))), ((((g1 . t0) * (h3 . t0)) * ( diff (f3,t0))) - (((g3 . t0) * (h1 . t0)) * ( diff (f1,t0)))), ((((g2 . t0) * (h1 . t0)) * ( diff (f1,t0))) - (((g1 . t0) * (h2 . t0)) * ( diff (f2,t0))))]|) + |[((((g3 . t0) * (f2 . t0)) * ( diff (h2,t0))) - (((g2 . t0) * (f3 . t0)) * ( diff (h3,t0)))), ((((g1 . t0) * (f3 . t0)) * ( diff (h3,t0))) - (((g3 . t0) * (f1 . t0)) * ( diff (h1,t0)))), ((((g2 . t0) * (f1 . t0)) * ( diff (h1,t0))) - (((g1 . t0) * (f2 . t0)) * ( diff (h2,t0))))]|) by RVSUM_1: 15;

      hence thesis;

    end;

    theorem :: EUCLID_8:114

    (f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0) & (g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0) & (h1 is_differentiable_in t0 & h2 is_differentiable_in t0 & h3 is_differentiable_in t0) implies ( VFuncdiff (((h2 (#) ((f1 (#) g2) - (f2 (#) g1))) - (h3 (#) ((f3 (#) g1) - (f1 (#) g3)))),((h3 (#) ((f2 (#) g3) - (f3 (#) g2))) - (h1 (#) ((f1 (#) g2) - (f2 (#) g1)))),((h1 (#) ((f3 (#) g1) - (f1 (#) g3))) - (h2 (#) ((f2 (#) g3) - (f3 (#) g2)))),t0)) = (( |[(((h2 . t0) * (((f1 . t0) * ( diff (g2,t0))) - ((f2 . t0) * ( diff (g1,t0))))) - ((h3 . t0) * (((f3 . t0) * ( diff (g1,t0))) - ((f1 . t0) * ( diff (g3,t0)))))), (((h3 . t0) * (((f2 . t0) * ( diff (g3,t0))) - ((f3 . t0) * ( diff (g2,t0))))) - ((h1 . t0) * (((f1 . t0) * ( diff (g2,t0))) - ((f2 . t0) * ( diff (g1,t0)))))), (((h1 . t0) * (((f3 . t0) * ( diff (g1,t0))) - ((f1 . t0) * ( diff (g3,t0))))) - ((h2 . t0) * (((f2 . t0) * ( diff (g3,t0))) - ((f3 . t0) * ( diff (g2,t0))))))]| + |[(((h2 . t0) * ((( diff (f1,t0)) * (g2 . t0)) - (( diff (f2,t0)) * (g1 . t0)))) - ((h3 . t0) * ((( diff (f3,t0)) * (g1 . t0)) - (( diff (f1,t0)) * (g3 . t0))))), (((h3 . t0) * ((( diff (f2,t0)) * (g3 . t0)) - (( diff (f3,t0)) * (g2 . t0)))) - ((h1 . t0) * ((( diff (f1,t0)) * (g2 . t0)) - (( diff (f2,t0)) * (g1 . t0))))), (((h1 . t0) * ((( diff (f3,t0)) * (g1 . t0)) - (( diff (f1,t0)) * (g3 . t0)))) - ((h2 . t0) * ((( diff (f2,t0)) * (g3 . t0)) - (( diff (f3,t0)) * (g2 . t0)))))]|) + |[((( diff (h2,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0)))) - (( diff (h3,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))))), ((( diff (h3,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))) - (( diff (h1,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))))), ((( diff (h1,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0)))) - (( diff (h2,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))))]|)

    proof

      assume that

       A1: f1 is_differentiable_in t0 & f2 is_differentiable_in t0 & f3 is_differentiable_in t0 and

       A2: g1 is_differentiable_in t0 & g2 is_differentiable_in t0 & g3 is_differentiable_in t0 and

       A3: h1 is_differentiable_in t0 & h2 is_differentiable_in t0 & h3 is_differentiable_in t0;

      

       A4: (f2 (#) g3) is_differentiable_in t0 & (f3 (#) g2) is_differentiable_in t0 & (f3 (#) g1) is_differentiable_in t0 & (f1 (#) g3) is_differentiable_in t0 & (f1 (#) g2) is_differentiable_in t0 & (f2 (#) g1) is_differentiable_in t0 by A1, A2, FDIFF_1: 16;

      then

       A5: ((f2 (#) g3) - (f3 (#) g2)) is_differentiable_in t0 & ((f3 (#) g1) - (f1 (#) g3)) is_differentiable_in t0 & ((f1 (#) g2) - (f2 (#) g1)) is_differentiable_in t0 by FDIFF_1: 14;

      then

       A6: (h3 (#) ((f2 (#) g3) - (f3 (#) g2))) is_differentiable_in t0 & (h3 (#) ((f3 (#) g1) - (f1 (#) g3))) is_differentiable_in t0 & (h2 (#) ((f1 (#) g2) - (f2 (#) g1))) is_differentiable_in t0 & (h2 (#) ((f2 (#) g3) - (f3 (#) g2))) is_differentiable_in t0 & (h1 (#) ((f3 (#) g1) - (f1 (#) g3))) is_differentiable_in t0 & (h1 (#) ((f1 (#) g2) - (f2 (#) g1))) is_differentiable_in t0 by A3, FDIFF_1: 16;

      

       A7: ((f1 (#) g2) - (f2 (#) g1)) is_left_differentiable_in t0 by A5, FDIFF_3: 22;

      

       A8: t0 in ( dom ((f1 (#) g2) - (f2 (#) g1)))

      proof

        consider r such that

         A9: 0 < r & [.(t0 - r), t0.] c= ( dom ((f1 (#) g2) - (f2 (#) g1))) by A7, FDIFF_3:def 4;

        (t0 - r) <= t0 by A9, XREAL_1: 44;

        then t0 in [.(t0 - r), t0.];

        hence thesis by A9;

      end;

      

       A10: ((f2 (#) g3) - (f3 (#) g2)) is_left_differentiable_in t0 by A5, FDIFF_3: 22;

      

       A11: t0 in ( dom ((f2 (#) g3) - (f3 (#) g2)))

      proof

        consider r1 such that

         A12: 0 < r1 & [.(t0 - r1), t0.] c= ( dom ((f2 (#) g3) - (f3 (#) g2))) by A10, FDIFF_3:def 4;

        (t0 - r1) <= t0 by A12, XREAL_1: 44;

        then t0 in [.(t0 - r1), t0.];

        hence thesis by A12;

      end;

      

       A13: ((f3 (#) g1) - (f1 (#) g3)) is_left_differentiable_in t0 by A5, FDIFF_3: 22;

      

       A14: t0 in ( dom ((f3 (#) g1) - (f1 (#) g3)))

      proof

        consider r2 such that

         A15: 0 < r2 & [.(t0 - r2), t0.] c= ( dom ((f3 (#) g1) - (f1 (#) g3))) by A13, FDIFF_3:def 4;

        (t0 - r2) <= t0 by A15, XREAL_1: 44;

        then t0 in [.(t0 - r2), t0.];

        hence thesis by A15;

      end;

      ( VFuncdiff (((h2 (#) ((f1 (#) g2) - (f2 (#) g1))) - (h3 (#) ((f3 (#) g1) - (f1 (#) g3)))),((h3 (#) ((f2 (#) g3) - (f3 (#) g2))) - (h1 (#) ((f1 (#) g2) - (f2 (#) g1)))),((h1 (#) ((f3 (#) g1) - (f1 (#) g3))) - (h2 (#) ((f2 (#) g3) - (f3 (#) g2)))),t0)) = (( VFuncdiff ((h2 (#) ((f1 (#) g2) - (f2 (#) g1))),(h3 (#) ((f2 (#) g3) - (f3 (#) g2))),(h1 (#) ((f3 (#) g1) - (f1 (#) g3))),t0)) - ( VFuncdiff ((h3 (#) ((f3 (#) g1) - (f1 (#) g3))),(h1 (#) ((f1 (#) g2) - (f2 (#) g1))),(h2 (#) ((f2 (#) g3) - (f3 (#) g2))),t0))) by A6, Th89

      .= (( |[((((f1 (#) g2) - (f2 (#) g1)) . t0) * ( diff (h2,t0))), ((((f2 (#) g3) - (f3 (#) g2)) . t0) * ( diff (h3,t0))), ((((f3 (#) g1) - (f1 (#) g3)) . t0) * ( diff (h1,t0)))]| + |[((h2 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h3 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0))), ((h1 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - ( VFuncdiff ((h3 (#) ((f3 (#) g1) - (f1 (#) g3))),(h1 (#) ((f1 (#) g2) - (f2 (#) g1))),(h2 (#) ((f2 (#) g3) - (f3 (#) g2))),t0))) by A3, A5, Th91

      .= (( |[((((f1 (#) g2) - (f2 (#) g1)) . t0) * ( diff (h2,t0))), ((((f2 (#) g3) - (f3 (#) g2)) . t0) * ( diff (h3,t0))), ((((f3 (#) g1) - (f1 (#) g3)) . t0) * ( diff (h1,t0)))]| + |[((h2 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h3 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0))), ((h1 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - ( |[((((f3 (#) g1) - (f1 (#) g3)) . t0) * ( diff (h3,t0))), ((((f1 (#) g2) - (f2 (#) g1)) . t0) * ( diff (h1,t0))), ((((f2 (#) g3) - (f3 (#) g2)) . t0) * ( diff (h2,t0)))]| + |[((h3 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0))), ((h1 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h2 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|)) by A3, A5, Th91

      .= (( |[((((f1 (#) g2) . t0) - ((f2 (#) g1) . t0)) * ( diff (h2,t0))), ((((f2 (#) g3) - (f3 (#) g2)) . t0) * ( diff (h3,t0))), ((((f3 (#) g1) - (f1 (#) g3)) . t0) * ( diff (h1,t0)))]| + |[((h2 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h3 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0))), ((h1 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - ( |[((((f3 (#) g1) - (f1 (#) g3)) . t0) * ( diff (h3,t0))), ((((f1 (#) g2) - (f2 (#) g1)) . t0) * ( diff (h1,t0))), ((((f2 (#) g3) - (f3 (#) g2)) . t0) * ( diff (h2,t0)))]| + |[((h3 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0))), ((h1 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h2 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|)) by A8, VALUED_1: 13

      .= (( |[((((f1 (#) g2) . t0) - ((f2 (#) g1) . t0)) * ( diff (h2,t0))), ((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * ( diff (h3,t0))), ((((f3 (#) g1) - (f1 (#) g3)) . t0) * ( diff (h1,t0)))]| + |[((h2 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h3 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0))), ((h1 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - ( |[((((f3 (#) g1) - (f1 (#) g3)) . t0) * ( diff (h3,t0))), ((((f1 (#) g2) - (f2 (#) g1)) . t0) * ( diff (h1,t0))), ((((f2 (#) g3) - (f3 (#) g2)) . t0) * ( diff (h2,t0)))]| + |[((h3 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0))), ((h1 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h2 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|)) by A11, VALUED_1: 13

      .= (( |[((((f1 (#) g2) . t0) - ((f2 (#) g1) . t0)) * ( diff (h2,t0))), ((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * ( diff (h3,t0))), ((((f3 (#) g1) . t0) - ((f1 (#) g3) . t0)) * ( diff (h1,t0)))]| + |[((h2 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h3 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0))), ((h1 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - ( |[((((f3 (#) g1) - (f1 (#) g3)) . t0) * ( diff (h3,t0))), ((((f1 (#) g2) - (f2 (#) g1)) . t0) * ( diff (h1,t0))), ((((f2 (#) g3) - (f3 (#) g2)) . t0) * ( diff (h2,t0)))]| + |[((h3 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0))), ((h1 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h2 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|)) by A14, VALUED_1: 13

      .= (( |[((((f1 (#) g2) . t0) - ((f2 (#) g1) . t0)) * ( diff (h2,t0))), ((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * ( diff (h3,t0))), ((((f3 (#) g1) . t0) - ((f1 (#) g3) . t0)) * ( diff (h1,t0)))]| + |[((h2 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h3 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0))), ((h1 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - ( |[((((f3 (#) g1) . t0) - ((f1 (#) g3) . t0)) * ( diff (h3,t0))), ((((f1 (#) g2) - (f2 (#) g1)) . t0) * ( diff (h1,t0))), ((((f2 (#) g3) - (f3 (#) g2)) . t0) * ( diff (h2,t0)))]| + |[((h3 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0))), ((h1 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h2 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|)) by A14, VALUED_1: 13

      .= (( |[((((f1 (#) g2) . t0) - ((f2 (#) g1) . t0)) * ( diff (h2,t0))), ((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * ( diff (h3,t0))), ((((f3 (#) g1) . t0) - ((f1 (#) g3) . t0)) * ( diff (h1,t0)))]| + |[((h2 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h3 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0))), ((h1 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - ( |[((((f3 (#) g1) . t0) - ((f1 (#) g3) . t0)) * ( diff (h3,t0))), ((((f1 (#) g2) . t0) - ((f2 (#) g1) . t0)) * ( diff (h1,t0))), ((((f2 (#) g3) - (f3 (#) g2)) . t0) * ( diff (h2,t0)))]| + |[((h3 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0))), ((h1 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h2 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|)) by A8, VALUED_1: 13

      .= (( |[((((f1 (#) g2) . t0) - ((f2 (#) g1) . t0)) * ( diff (h2,t0))), ((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * ( diff (h3,t0))), ((((f3 (#) g1) . t0) - ((f1 (#) g3) . t0)) * ( diff (h1,t0)))]| + |[((h2 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h3 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0))), ((h1 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - ( |[((((f3 (#) g1) . t0) - ((f1 (#) g3) . t0)) * ( diff (h3,t0))), ((((f1 (#) g2) . t0) - ((f2 (#) g1) . t0)) * ( diff (h1,t0))), ((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * ( diff (h2,t0)))]| + |[((h3 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0))), ((h1 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h2 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|)) by A11, VALUED_1: 13

      .= (( |[((((f1 . t0) * (g2 . t0)) - ((f2 (#) g1) . t0)) * ( diff (h2,t0))), ((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * ( diff (h3,t0))), ((((f3 (#) g1) . t0) - ((f1 (#) g3) . t0)) * ( diff (h1,t0)))]| + |[((h2 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h3 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0))), ((h1 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - ( |[((((f3 (#) g1) . t0) - ((f1 (#) g3) . t0)) * ( diff (h3,t0))), ((((f1 (#) g2) . t0) - ((f2 (#) g1) . t0)) * ( diff (h1,t0))), ((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * ( diff (h2,t0)))]| + |[((h3 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0))), ((h1 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h2 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|)) by VALUED_1: 5

      .= (( |[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h2,t0))), ((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * ( diff (h3,t0))), ((((f3 (#) g1) . t0) - ((f1 (#) g3) . t0)) * ( diff (h1,t0)))]| + |[((h2 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h3 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0))), ((h1 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - ( |[((((f3 (#) g1) . t0) - ((f1 (#) g3) . t0)) * ( diff (h3,t0))), ((((f1 (#) g2) . t0) - ((f2 (#) g1) . t0)) * ( diff (h1,t0))), ((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * ( diff (h2,t0)))]| + |[((h3 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0))), ((h1 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h2 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|)) by VALUED_1: 5

      .= (( |[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h2,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 (#) g2) . t0)) * ( diff (h3,t0))), ((((f3 (#) g1) . t0) - ((f1 (#) g3) . t0)) * ( diff (h1,t0)))]| + |[((h2 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h3 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0))), ((h1 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - ( |[((((f3 (#) g1) . t0) - ((f1 (#) g3) . t0)) * ( diff (h3,t0))), ((((f1 (#) g2) . t0) - ((f2 (#) g1) . t0)) * ( diff (h1,t0))), ((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * ( diff (h2,t0)))]| + |[((h3 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0))), ((h1 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h2 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|)) by VALUED_1: 5

      .= (( |[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h2,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h3,t0))), ((((f3 (#) g1) . t0) - ((f1 (#) g3) . t0)) * ( diff (h1,t0)))]| + |[((h2 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h3 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0))), ((h1 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - ( |[((((f3 (#) g1) . t0) - ((f1 (#) g3) . t0)) * ( diff (h3,t0))), ((((f1 (#) g2) . t0) - ((f2 (#) g1) . t0)) * ( diff (h1,t0))), ((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * ( diff (h2,t0)))]| + |[((h3 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0))), ((h1 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h2 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|)) by VALUED_1: 5

      .= (( |[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h2,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h3,t0))), ((((f3 . t0) * (g1 . t0)) - ((f1 (#) g3) . t0)) * ( diff (h1,t0)))]| + |[((h2 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h3 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0))), ((h1 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - ( |[((((f3 (#) g1) . t0) - ((f1 (#) g3) . t0)) * ( diff (h3,t0))), ((((f1 (#) g2) . t0) - ((f2 (#) g1) . t0)) * ( diff (h1,t0))), ((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * ( diff (h2,t0)))]| + |[((h3 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0))), ((h1 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h2 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|)) by VALUED_1: 5

      .= (( |[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h2,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h3,t0))), ((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h1,t0)))]| + |[((h2 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h3 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0))), ((h1 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - ( |[((((f3 (#) g1) . t0) - ((f1 (#) g3) . t0)) * ( diff (h3,t0))), ((((f1 (#) g2) . t0) - ((f2 (#) g1) . t0)) * ( diff (h1,t0))), ((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * ( diff (h2,t0)))]| + |[((h3 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0))), ((h1 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h2 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|)) by VALUED_1: 5

      .= (( |[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h2,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h3,t0))), ((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h1,t0)))]| + |[((h2 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h3 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0))), ((h1 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - ( |[((((f3 . t0) * (g1 . t0)) - ((f1 (#) g3) . t0)) * ( diff (h3,t0))), ((((f1 (#) g2) . t0) - ((f2 (#) g1) . t0)) * ( diff (h1,t0))), ((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * ( diff (h2,t0)))]| + |[((h3 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0))), ((h1 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h2 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|)) by VALUED_1: 5

      .= (( |[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h2,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h3,t0))), ((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h1,t0)))]| + |[((h2 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h3 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0))), ((h1 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - ( |[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h3,t0))), ((((f1 (#) g2) . t0) - ((f2 (#) g1) . t0)) * ( diff (h1,t0))), ((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * ( diff (h2,t0)))]| + |[((h3 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0))), ((h1 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h2 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|)) by VALUED_1: 5

      .= (( |[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h2,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h3,t0))), ((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h1,t0)))]| + |[((h2 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h3 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0))), ((h1 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - ( |[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h3,t0))), ((((f1 . t0) * (g2 . t0)) - ((f2 (#) g1) . t0)) * ( diff (h1,t0))), ((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * ( diff (h2,t0)))]| + |[((h3 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0))), ((h1 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h2 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|)) by VALUED_1: 5

      .= (( |[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h2,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h3,t0))), ((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h1,t0)))]| + |[((h2 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h3 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0))), ((h1 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - ( |[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h3,t0))), ((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h1,t0))), ((((f2 (#) g3) . t0) - ((f3 (#) g2) . t0)) * ( diff (h2,t0)))]| + |[((h3 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0))), ((h1 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h2 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|)) by VALUED_1: 5

      .= (( |[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h2,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h3,t0))), ((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h1,t0)))]| + |[((h2 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h3 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0))), ((h1 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - ( |[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h3,t0))), ((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h1,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 (#) g2) . t0)) * ( diff (h2,t0)))]| + |[((h3 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0))), ((h1 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h2 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|)) by VALUED_1: 5

      .= (( |[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h2,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h3,t0))), ((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h1,t0)))]| + |[((h2 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h3 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0))), ((h1 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - ( |[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h3,t0))), ((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h1,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h2,t0)))]| + |[((h3 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0))), ((h1 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h2 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|)) by VALUED_1: 5

      .= (( |[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h2,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h3,t0))), ((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h1,t0)))]| + |[((h2 . t0) * (( diff ((f1 (#) g2),t0)) - ( diff ((f2 (#) g1),t0)))), ((h3 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0))), ((h1 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - ( |[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h3,t0))), ((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h1,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h2,t0)))]| + |[((h3 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0))), ((h1 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h2 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|)) by A4, FDIFF_1: 14

      .= (( |[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h2,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h3,t0))), ((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h1,t0)))]| + |[((h2 . t0) * (( diff ((f1 (#) g2),t0)) - ( diff ((f2 (#) g1),t0)))), ((h3 . t0) * (( diff ((f2 (#) g3),t0)) - ( diff ((f3 (#) g2),t0)))), ((h1 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0)))]|) - ( |[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h3,t0))), ((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h1,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h2,t0)))]| + |[((h3 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0))), ((h1 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h2 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|)) by A4, FDIFF_1: 14

      .= (( |[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h2,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h3,t0))), ((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h1,t0)))]| + |[((h2 . t0) * (( diff ((f1 (#) g2),t0)) - ( diff ((f2 (#) g1),t0)))), ((h3 . t0) * (( diff ((f2 (#) g3),t0)) - ( diff ((f3 (#) g2),t0)))), ((h1 . t0) * (( diff ((f3 (#) g1),t0)) - ( diff ((f1 (#) g3),t0))))]|) - ( |[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h3,t0))), ((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h1,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h2,t0)))]| + |[((h3 . t0) * ( diff (((f3 (#) g1) - (f1 (#) g3)),t0))), ((h1 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h2 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|)) by A4, FDIFF_1: 14

      .= (( |[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h2,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h3,t0))), ((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h1,t0)))]| + |[((h2 . t0) * (( diff ((f1 (#) g2),t0)) - ( diff ((f2 (#) g1),t0)))), ((h3 . t0) * (( diff ((f2 (#) g3),t0)) - ( diff ((f3 (#) g2),t0)))), ((h1 . t0) * (( diff ((f3 (#) g1),t0)) - ( diff ((f1 (#) g3),t0))))]|) - ( |[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h3,t0))), ((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h1,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h2,t0)))]| + |[((h3 . t0) * (( diff ((f3 (#) g1),t0)) - ( diff ((f1 (#) g3),t0)))), ((h1 . t0) * ( diff (((f1 (#) g2) - (f2 (#) g1)),t0))), ((h2 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|)) by A4, FDIFF_1: 14

      .= (( |[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h2,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h3,t0))), ((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h1,t0)))]| + |[((h2 . t0) * (( diff ((f1 (#) g2),t0)) - ( diff ((f2 (#) g1),t0)))), ((h3 . t0) * (( diff ((f2 (#) g3),t0)) - ( diff ((f3 (#) g2),t0)))), ((h1 . t0) * (( diff ((f3 (#) g1),t0)) - ( diff ((f1 (#) g3),t0))))]|) - ( |[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h3,t0))), ((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h1,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h2,t0)))]| + |[((h3 . t0) * (( diff ((f3 (#) g1),t0)) - ( diff ((f1 (#) g3),t0)))), ((h1 . t0) * (( diff ((f1 (#) g2),t0)) - ( diff ((f2 (#) g1),t0)))), ((h2 . t0) * ( diff (((f2 (#) g3) - (f3 (#) g2)),t0)))]|)) by A4, FDIFF_1: 14

      .= (( |[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h2,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h3,t0))), ((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h1,t0)))]| + |[((h2 . t0) * (( diff ((f1 (#) g2),t0)) - ( diff ((f2 (#) g1),t0)))), ((h3 . t0) * (( diff ((f2 (#) g3),t0)) - ( diff ((f3 (#) g2),t0)))), ((h1 . t0) * (( diff ((f3 (#) g1),t0)) - ( diff ((f1 (#) g3),t0))))]|) - ( |[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h3,t0))), ((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h1,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h2,t0)))]| + |[((h3 . t0) * (( diff ((f3 (#) g1),t0)) - ( diff ((f1 (#) g3),t0)))), ((h1 . t0) * (( diff ((f1 (#) g2),t0)) - ( diff ((f2 (#) g1),t0)))), ((h2 . t0) * (( diff ((f2 (#) g3),t0)) - ( diff ((f3 (#) g2),t0))))]|)) by A4, FDIFF_1: 14

      .= (( |[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h2,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h3,t0))), ((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h1,t0)))]| + |[((h2 . t0) * ((((g2 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g2,t0)))) - ( diff ((f2 (#) g1),t0)))), ((h3 . t0) * (( diff ((f2 (#) g3),t0)) - ( diff ((f3 (#) g2),t0)))), ((h1 . t0) * (( diff ((f3 (#) g1),t0)) - ( diff ((f1 (#) g3),t0))))]|) - ( |[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h3,t0))), ((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h1,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h2,t0)))]| + |[((h3 . t0) * (( diff ((f3 (#) g1),t0)) - ( diff ((f1 (#) g3),t0)))), ((h1 . t0) * (( diff ((f1 (#) g2),t0)) - ( diff ((f2 (#) g1),t0)))), ((h2 . t0) * (( diff ((f2 (#) g3),t0)) - ( diff ((f3 (#) g2),t0))))]|)) by A1, A2, FDIFF_1: 16

      .= (( |[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h2,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h3,t0))), ((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h1,t0)))]| + |[((h2 . t0) * ((((g2 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g2,t0)))) - (((g1 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g1,t0)))))), ((h3 . t0) * (( diff ((f2 (#) g3),t0)) - ( diff ((f3 (#) g2),t0)))), ((h1 . t0) * (( diff ((f3 (#) g1),t0)) - ( diff ((f1 (#) g3),t0))))]|) - ( |[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h3,t0))), ((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h1,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h2,t0)))]| + |[((h3 . t0) * (( diff ((f3 (#) g1),t0)) - ( diff ((f1 (#) g3),t0)))), ((h1 . t0) * (( diff ((f1 (#) g2),t0)) - ( diff ((f2 (#) g1),t0)))), ((h2 . t0) * (( diff ((f2 (#) g3),t0)) - ( diff ((f3 (#) g2),t0))))]|)) by A1, A2, FDIFF_1: 16

      .= (( |[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h2,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h3,t0))), ((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h1,t0)))]| + |[((h2 . t0) * ((((g2 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g2,t0)))) - (((g1 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g1,t0)))))), ((h3 . t0) * ((((g3 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g3,t0)))) - ( diff ((f3 (#) g2),t0)))), ((h1 . t0) * (( diff ((f3 (#) g1),t0)) - ( diff ((f1 (#) g3),t0))))]|) - ( |[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h3,t0))), ((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h1,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h2,t0)))]| + |[((h3 . t0) * (( diff ((f3 (#) g1),t0)) - ( diff ((f1 (#) g3),t0)))), ((h1 . t0) * (( diff ((f1 (#) g2),t0)) - ( diff ((f2 (#) g1),t0)))), ((h2 . t0) * (( diff ((f2 (#) g3),t0)) - ( diff ((f3 (#) g2),t0))))]|)) by A1, A2, FDIFF_1: 16

      .= (( |[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h2,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h3,t0))), ((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h1,t0)))]| + |[((h2 . t0) * ((((g2 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g2,t0)))) - (((g1 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g1,t0)))))), ((h3 . t0) * ((((g3 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g3,t0)))) - (((g2 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g2,t0)))))), ((h1 . t0) * (( diff ((f3 (#) g1),t0)) - ( diff ((f1 (#) g3),t0))))]|) - ( |[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h3,t0))), ((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h1,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h2,t0)))]| + |[((h3 . t0) * (( diff ((f3 (#) g1),t0)) - ( diff ((f1 (#) g3),t0)))), ((h1 . t0) * (( diff ((f1 (#) g2),t0)) - ( diff ((f2 (#) g1),t0)))), ((h2 . t0) * (( diff ((f2 (#) g3),t0)) - ( diff ((f3 (#) g2),t0))))]|)) by A1, A2, FDIFF_1: 16

      .= (( |[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h2,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h3,t0))), ((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h1,t0)))]| + |[((h2 . t0) * ((((g2 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g2,t0)))) - (((g1 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g1,t0)))))), ((h3 . t0) * ((((g3 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g3,t0)))) - (((g2 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g2,t0)))))), ((h1 . t0) * ((((g1 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g1,t0)))) - ( diff ((f1 (#) g3),t0))))]|) - ( |[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h3,t0))), ((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h1,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h2,t0)))]| + |[((h3 . t0) * (( diff ((f3 (#) g1),t0)) - ( diff ((f1 (#) g3),t0)))), ((h1 . t0) * (( diff ((f1 (#) g2),t0)) - ( diff ((f2 (#) g1),t0)))), ((h2 . t0) * (( diff ((f2 (#) g3),t0)) - ( diff ((f3 (#) g2),t0))))]|)) by A1, A2, FDIFF_1: 16

      .= (( |[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h2,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h3,t0))), ((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h1,t0)))]| + |[((h2 . t0) * ((((g2 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g2,t0)))) - (((g1 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g1,t0)))))), ((h3 . t0) * ((((g3 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g3,t0)))) - (((g2 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g2,t0)))))), ((h1 . t0) * ((((g1 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g1,t0)))) - (((g3 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g3,t0))))))]|) - ( |[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h3,t0))), ((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h1,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h2,t0)))]| + |[((h3 . t0) * (( diff ((f3 (#) g1),t0)) - ( diff ((f1 (#) g3),t0)))), ((h1 . t0) * (( diff ((f1 (#) g2),t0)) - ( diff ((f2 (#) g1),t0)))), ((h2 . t0) * (( diff ((f2 (#) g3),t0)) - ( diff ((f3 (#) g2),t0))))]|)) by A1, A2, FDIFF_1: 16

      .= (( |[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h2,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h3,t0))), ((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h1,t0)))]| + |[((h2 . t0) * ((((g2 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g2,t0)))) - (((g1 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g1,t0)))))), ((h3 . t0) * ((((g3 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g3,t0)))) - (((g2 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g2,t0)))))), ((h1 . t0) * ((((g1 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g1,t0)))) - (((g3 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g3,t0))))))]|) - ( |[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h3,t0))), ((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h1,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h2,t0)))]| + |[((h3 . t0) * ((((g1 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g1,t0)))) - ( diff ((f1 (#) g3),t0)))), ((h1 . t0) * (( diff ((f1 (#) g2),t0)) - ( diff ((f2 (#) g1),t0)))), ((h2 . t0) * (( diff ((f2 (#) g3),t0)) - ( diff ((f3 (#) g2),t0))))]|)) by A1, A2, FDIFF_1: 16

      .= (( |[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h2,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h3,t0))), ((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h1,t0)))]| + |[((h2 . t0) * ((((g2 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g2,t0)))) - (((g1 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g1,t0)))))), ((h3 . t0) * ((((g3 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g3,t0)))) - (((g2 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g2,t0)))))), ((h1 . t0) * ((((g1 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g1,t0)))) - (((g3 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g3,t0))))))]|) - ( |[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h3,t0))), ((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h1,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h2,t0)))]| + |[((h3 . t0) * ((((g1 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g1,t0)))) - (((g3 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g3,t0)))))), ((h1 . t0) * (( diff ((f1 (#) g2),t0)) - ( diff ((f2 (#) g1),t0)))), ((h2 . t0) * (( diff ((f2 (#) g3),t0)) - ( diff ((f3 (#) g2),t0))))]|)) by A1, A2, FDIFF_1: 16

      .= (( |[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h2,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h3,t0))), ((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h1,t0)))]| + |[((h2 . t0) * ((((g2 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g2,t0)))) - (((g1 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g1,t0)))))), ((h3 . t0) * ((((g3 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g3,t0)))) - (((g2 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g2,t0)))))), ((h1 . t0) * ((((g1 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g1,t0)))) - (((g3 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g3,t0))))))]|) - ( |[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h3,t0))), ((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h1,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h2,t0)))]| + |[((h3 . t0) * ((((g1 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g1,t0)))) - (((g3 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g3,t0)))))), ((h1 . t0) * ((((g2 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g2,t0)))) - ( diff ((f2 (#) g1),t0)))), ((h2 . t0) * (( diff ((f2 (#) g3),t0)) - ( diff ((f3 (#) g2),t0))))]|)) by A1, A2, FDIFF_1: 16

      .= (( |[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h2,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h3,t0))), ((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h1,t0)))]| + |[((h2 . t0) * ((((g2 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g2,t0)))) - (((g1 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g1,t0)))))), ((h3 . t0) * ((((g3 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g3,t0)))) - (((g2 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g2,t0)))))), ((h1 . t0) * ((((g1 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g1,t0)))) - (((g3 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g3,t0))))))]|) - ( |[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h3,t0))), ((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h1,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h2,t0)))]| + |[((h3 . t0) * ((((g1 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g1,t0)))) - (((g3 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g3,t0)))))), ((h1 . t0) * ((((g2 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g2,t0)))) - (((g1 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g1,t0)))))), ((h2 . t0) * (( diff ((f2 (#) g3),t0)) - ( diff ((f3 (#) g2),t0))))]|)) by A1, A2, FDIFF_1: 16

      .= (( |[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h2,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h3,t0))), ((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h1,t0)))]| + |[((h2 . t0) * ((((g2 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g2,t0)))) - (((g1 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g1,t0)))))), ((h3 . t0) * ((((g3 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g3,t0)))) - (((g2 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g2,t0)))))), ((h1 . t0) * ((((g1 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g1,t0)))) - (((g3 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g3,t0))))))]|) - ( |[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h3,t0))), ((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h1,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h2,t0)))]| + |[((h3 . t0) * ((((g1 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g1,t0)))) - (((g3 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g3,t0)))))), ((h1 . t0) * ((((g2 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g2,t0)))) - (((g1 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g1,t0)))))), ((h2 . t0) * ((((g3 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g3,t0)))) - ( diff ((f3 (#) g2),t0))))]|)) by A1, A2, FDIFF_1: 16

      .= (( |[((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h2,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h3,t0))), ((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h1,t0)))]| + |[((h2 . t0) * ((((g2 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g2,t0)))) - (((g1 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g1,t0)))))), ((h3 . t0) * ((((g3 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g3,t0)))) - (((g2 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g2,t0)))))), ((h1 . t0) * ((((g1 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g1,t0)))) - (((g3 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g3,t0))))))]|) - ( |[((((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))) * ( diff (h3,t0))), ((((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))) * ( diff (h1,t0))), ((((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))) * ( diff (h2,t0)))]| + |[((h3 . t0) * ((((g1 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g1,t0)))) - (((g3 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g3,t0)))))), ((h1 . t0) * ((((g2 . t0) * ( diff (f1,t0))) + ((f1 . t0) * ( diff (g2,t0)))) - (((g1 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g1,t0)))))), ((h2 . t0) * ((((g3 . t0) * ( diff (f2,t0))) + ((f2 . t0) * ( diff (g3,t0)))) - (((g2 . t0) * ( diff (f3,t0))) + ((f3 . t0) * ( diff (g2,t0))))))]|)) by A1, A2, FDIFF_1: 16

      .= ((( |[(((h2 . t0) * (((f1 . t0) * ( diff (g2,t0))) - ((f2 . t0) * ( diff (g1,t0))))) + ((h2 . t0) * ((( diff (f1,t0)) * (g2 . t0)) - (( diff (f2,t0)) * (g1 . t0))))), (((h3 . t0) * (((f2 . t0) * ( diff (g3,t0))) - ((f3 . t0) * ( diff (g2,t0))))) + ((h3 . t0) * ((( diff (f2,t0)) * (g3 . t0)) - (( diff (f3,t0)) * (g2 . t0))))), (((h1 . t0) * (((f3 . t0) * ( diff (g1,t0))) - ((f1 . t0) * ( diff (g3,t0))))) + ((h1 . t0) * ((( diff (f3,t0)) * (g1 . t0)) - (( diff (f1,t0)) * (g3 . t0)))))]| + |[(( diff (h2,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0)))), (( diff (h3,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))), (( diff (h1,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))))]|) - |[(( diff (h3,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0)))), (( diff (h1,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0)))), (( diff (h2,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))))]|) - |[(((h3 . t0) * (((f3 . t0) * ( diff (g1,t0))) - ((f1 . t0) * ( diff (g3,t0))))) + ((h3 . t0) * ((( diff (f3,t0)) * (g1 . t0)) - (( diff (f1,t0)) * (g3 . t0))))), (((h1 . t0) * (((f1 . t0) * ( diff (g2,t0))) - ((f2 . t0) * ( diff (g1,t0))))) + ((h1 . t0) * ((( diff (f1,t0)) * (g2 . t0)) - (( diff (f2,t0)) * (g1 . t0))))), (((h2 . t0) * (((g3 . t0) * ( diff (f2,t0))) - ((g2 . t0) * ( diff (f3,t0))))) + ((h2 . t0) * (((f2 . t0) * ( diff (g3,t0))) - ((f3 . t0) * ( diff (g2,t0))))))]|) by RVSUM_1: 39

      .= (( |[(((h2 . t0) * (((f1 . t0) * ( diff (g2,t0))) - ((f2 . t0) * ( diff (g1,t0))))) + ((h2 . t0) * ((( diff (f1,t0)) * (g2 . t0)) - (( diff (f2,t0)) * (g1 . t0))))), (((h3 . t0) * (((f2 . t0) * ( diff (g3,t0))) - ((f3 . t0) * ( diff (g2,t0))))) + ((h3 . t0) * ((( diff (f2,t0)) * (g3 . t0)) - (( diff (f3,t0)) * (g2 . t0))))), (((h1 . t0) * (((f3 . t0) * ( diff (g1,t0))) - ((f1 . t0) * ( diff (g3,t0))))) + ((h1 . t0) * ((( diff (f3,t0)) * (g1 . t0)) - (( diff (f1,t0)) * (g3 . t0)))))]| + ( |[(( diff (h2,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0)))), (( diff (h3,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))), (( diff (h1,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))))]| - |[(( diff (h3,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0)))), (( diff (h1,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0)))), (( diff (h2,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0))))]|)) - |[(((h3 . t0) * (((f3 . t0) * ( diff (g1,t0))) - ((f1 . t0) * ( diff (g3,t0))))) + ((h3 . t0) * ((( diff (f3,t0)) * (g1 . t0)) - (( diff (f1,t0)) * (g3 . t0))))), (((h1 . t0) * (((f1 . t0) * ( diff (g2,t0))) - ((f2 . t0) * ( diff (g1,t0))))) + ((h1 . t0) * ((( diff (f1,t0)) * (g2 . t0)) - (( diff (f2,t0)) * (g1 . t0))))), (((h2 . t0) * (((g3 . t0) * ( diff (f2,t0))) - ((g2 . t0) * ( diff (f3,t0))))) + ((h2 . t0) * (((f2 . t0) * ( diff (g3,t0))) - ((f3 . t0) * ( diff (g2,t0))))))]|) by RVSUM_1: 15

      .= (( |[(((h2 . t0) * (((f1 . t0) * ( diff (g2,t0))) - ((f2 . t0) * ( diff (g1,t0))))) + ((h2 . t0) * ((( diff (f1,t0)) * (g2 . t0)) - (( diff (f2,t0)) * (g1 . t0))))), (((h3 . t0) * (((f2 . t0) * ( diff (g3,t0))) - ((f3 . t0) * ( diff (g2,t0))))) + ((h3 . t0) * ((( diff (f2,t0)) * (g3 . t0)) - (( diff (f3,t0)) * (g2 . t0))))), (((h1 . t0) * (((f3 . t0) * ( diff (g1,t0))) - ((f1 . t0) * ( diff (g3,t0))))) + ((h1 . t0) * ((( diff (f3,t0)) * (g1 . t0)) - (( diff (f1,t0)) * (g3 . t0)))))]| + |[((( diff (h2,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0)))) - (( diff (h3,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))))), ((( diff (h3,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))) - (( diff (h1,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))))), ((( diff (h1,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0)))) - (( diff (h2,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))))]|) - |[(((h3 . t0) * (((f3 . t0) * ( diff (g1,t0))) - ((f1 . t0) * ( diff (g3,t0))))) + ((h3 . t0) * ((( diff (f3,t0)) * (g1 . t0)) - (( diff (f1,t0)) * (g3 . t0))))), (((h1 . t0) * (((f1 . t0) * ( diff (g2,t0))) - ((f2 . t0) * ( diff (g1,t0))))) + ((h1 . t0) * ((( diff (f1,t0)) * (g2 . t0)) - (( diff (f2,t0)) * (g1 . t0))))), (((h2 . t0) * (((g3 . t0) * ( diff (f2,t0))) - ((g2 . t0) * ( diff (f3,t0))))) + ((h2 . t0) * (((f2 . t0) * ( diff (g3,t0))) - ((f3 . t0) * ( diff (g2,t0))))))]|) by Lm11

      .= (( |[(((h2 . t0) * (((f1 . t0) * ( diff (g2,t0))) - ((f2 . t0) * ( diff (g1,t0))))) + ((h2 . t0) * ((( diff (f1,t0)) * (g2 . t0)) - (( diff (f2,t0)) * (g1 . t0))))), (((h3 . t0) * (((f2 . t0) * ( diff (g3,t0))) - ((f3 . t0) * ( diff (g2,t0))))) + ((h3 . t0) * ((( diff (f2,t0)) * (g3 . t0)) - (( diff (f3,t0)) * (g2 . t0))))), (((h1 . t0) * (((f3 . t0) * ( diff (g1,t0))) - ((f1 . t0) * ( diff (g3,t0))))) + ((h1 . t0) * ((( diff (f3,t0)) * (g1 . t0)) - (( diff (f1,t0)) * (g3 . t0)))))]| - |[(((h3 . t0) * (((f3 . t0) * ( diff (g1,t0))) - ((f1 . t0) * ( diff (g3,t0))))) + ((h3 . t0) * ((( diff (f3,t0)) * (g1 . t0)) - (( diff (f1,t0)) * (g3 . t0))))), (((h1 . t0) * (((f1 . t0) * ( diff (g2,t0))) - ((f2 . t0) * ( diff (g1,t0))))) + ((h1 . t0) * ((( diff (f1,t0)) * (g2 . t0)) - (( diff (f2,t0)) * (g1 . t0))))), (((h2 . t0) * (((g3 . t0) * ( diff (f2,t0))) - ((g2 . t0) * ( diff (f3,t0))))) + ((h2 . t0) * (((f2 . t0) * ( diff (g3,t0))) - ((f3 . t0) * ( diff (g2,t0))))))]|) + |[((( diff (h2,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0)))) - (( diff (h3,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))))), ((( diff (h3,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))) - (( diff (h1,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))))), ((( diff (h1,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0)))) - (( diff (h2,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))))]|) by RVSUM_1: 15

      .= ( |[((((h2 . t0) * (((f1 . t0) * ( diff (g2,t0))) - ((f2 . t0) * ( diff (g1,t0))))) + ((h2 . t0) * ((( diff (f1,t0)) * (g2 . t0)) - (( diff (f2,t0)) * (g1 . t0))))) - (((h3 . t0) * (((f3 . t0) * ( diff (g1,t0))) - ((f1 . t0) * ( diff (g3,t0))))) + ((h3 . t0) * ((( diff (f3,t0)) * (g1 . t0)) - (( diff (f1,t0)) * (g3 . t0)))))), ((((h3 . t0) * (((f2 . t0) * ( diff (g3,t0))) - ((f3 . t0) * ( diff (g2,t0))))) + ((h3 . t0) * ((( diff (f2,t0)) * (g3 . t0)) - (( diff (f3,t0)) * (g2 . t0))))) - (((h1 . t0) * (((f1 . t0) * ( diff (g2,t0))) - ((f2 . t0) * ( diff (g1,t0))))) + ((h1 . t0) * ((( diff (f1,t0)) * (g2 . t0)) - (( diff (f2,t0)) * (g1 . t0)))))), ((((h1 . t0) * (((f3 . t0) * ( diff (g1,t0))) - ((f1 . t0) * ( diff (g3,t0))))) + ((h1 . t0) * ((( diff (f3,t0)) * (g1 . t0)) - (( diff (f1,t0)) * (g3 . t0))))) - (((h2 . t0) * (((g3 . t0) * ( diff (f2,t0))) - ((g2 . t0) * ( diff (f3,t0))))) + ((h2 . t0) * (((f2 . t0) * ( diff (g3,t0))) - ((f3 . t0) * ( diff (g2,t0)))))))]| + |[((( diff (h2,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0)))) - (( diff (h3,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))))), ((( diff (h3,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))) - (( diff (h1,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))))), ((( diff (h1,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0)))) - (( diff (h2,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))))]|) by Lm11

      .= ( |[((((h2 . t0) * (((f1 . t0) * ( diff (g2,t0))) - ((f2 . t0) * ( diff (g1,t0))))) - ((h3 . t0) * (((f3 . t0) * ( diff (g1,t0))) - ((f1 . t0) * ( diff (g3,t0)))))) + (((h2 . t0) * ((( diff (f1,t0)) * (g2 . t0)) - (( diff (f2,t0)) * (g1 . t0)))) - ((h3 . t0) * ((( diff (f3,t0)) * (g1 . t0)) - (( diff (f1,t0)) * (g3 . t0)))))), ((((h3 . t0) * (((f2 . t0) * ( diff (g3,t0))) - ((f3 . t0) * ( diff (g2,t0))))) - ((h1 . t0) * (((f1 . t0) * ( diff (g2,t0))) - ((f2 . t0) * ( diff (g1,t0)))))) + (((h3 . t0) * ((( diff (f2,t0)) * (g3 . t0)) - (( diff (f3,t0)) * (g2 . t0)))) - ((h1 . t0) * ((( diff (f1,t0)) * (g2 . t0)) - (( diff (f2,t0)) * (g1 . t0)))))), ((((h1 . t0) * (((f3 . t0) * ( diff (g1,t0))) - ((f1 . t0) * ( diff (g3,t0))))) - ((h2 . t0) * (((f2 . t0) * ( diff (g3,t0))) - ((f3 . t0) * ( diff (g2,t0)))))) + (((h1 . t0) * ((( diff (f3,t0)) * (g1 . t0)) - (( diff (f1,t0)) * (g3 . t0)))) - ((h2 . t0) * (((g3 . t0) * ( diff (f2,t0))) - ((g2 . t0) * ( diff (f3,t0)))))))]| + |[((( diff (h2,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0)))) - (( diff (h3,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))))), ((( diff (h3,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))) - (( diff (h1,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))))), ((( diff (h1,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0)))) - (( diff (h2,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))))]|)

      .= (( |[(((h2 . t0) * (((f1 . t0) * ( diff (g2,t0))) - ((f2 . t0) * ( diff (g1,t0))))) - ((h3 . t0) * (((f3 . t0) * ( diff (g1,t0))) - ((f1 . t0) * ( diff (g3,t0)))))), (((h3 . t0) * (((f2 . t0) * ( diff (g3,t0))) - ((f3 . t0) * ( diff (g2,t0))))) - ((h1 . t0) * (((f1 . t0) * ( diff (g2,t0))) - ((f2 . t0) * ( diff (g1,t0)))))), (((h1 . t0) * (((f3 . t0) * ( diff (g1,t0))) - ((f1 . t0) * ( diff (g3,t0))))) - ((h2 . t0) * (((f2 . t0) * ( diff (g3,t0))) - ((f3 . t0) * ( diff (g2,t0))))))]| + |[(((h2 . t0) * ((( diff (f1,t0)) * (g2 . t0)) - (( diff (f2,t0)) * (g1 . t0)))) - ((h3 . t0) * ((( diff (f3,t0)) * (g1 . t0)) - (( diff (f1,t0)) * (g3 . t0))))), (((h3 . t0) * ((( diff (f2,t0)) * (g3 . t0)) - (( diff (f3,t0)) * (g2 . t0)))) - ((h1 . t0) * ((( diff (f1,t0)) * (g2 . t0)) - (( diff (f2,t0)) * (g1 . t0))))), (((h1 . t0) * ((( diff (f3,t0)) * (g1 . t0)) - (( diff (f1,t0)) * (g3 . t0)))) - ((h2 . t0) * ((( diff (f2,t0)) * (g3 . t0)) - (( diff (f3,t0)) * (g2 . t0)))))]|) + |[((( diff (h2,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0)))) - (( diff (h3,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0))))), ((( diff (h3,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))) - (( diff (h1,t0)) * (((f1 . t0) * (g2 . t0)) - ((f2 . t0) * (g1 . t0))))), ((( diff (h1,t0)) * (((f3 . t0) * (g1 . t0)) - ((f1 . t0) * (g3 . t0)))) - (( diff (h2,t0)) * (((f2 . t0) * (g3 . t0)) - ((f3 . t0) * (g2 . t0)))))]|) by Lm8;

      hence thesis;

    end;