reserve x for set,
  i,j,k,n for Nat,
  K for Field;
reserve a,b,c,d for Element of K;
reserve D for non empty set;

theorem Th37:
  <*2,1,3*> * <*1,3,2*> = <*2,3,1*> & <*1,3,2*> * <*2,1,3*> = <*3,
1,2*> & <*2,1,3*> * <*3,2,1*> = <*3,1,2*> & <*3,2,1*> * <*2,1,3*> = <*2,3,1*> &
<*3,2,1*> * <*3,2,1*> = <*1,2,3*> & <*2,1,3*> * <*2,1,3*> = <*1,2,3*> & <*1,3,2
*> * <*1,3,2*> = <*1,2,3*> & <*1,3,2*> * <*2,3,1*> = <*3,2,1*> & <*2,3,1*> * <*
2,3,1*> = <*3,1,2*> & <*2,3,1*> * <*3,1,2*> = <*1,2,3*> & <*3,1,2*> * <*2,3,1*>
= <*1,2,3*> & <*3,1,2*> * <*3,1,2*> = <*2,3,1*> & <*1,3,2*> * <*3,2,1*> = <*2,3
  ,1*> & <*3,2,1*> * <*1,3,2*> = <*3,1,2*>
proof
  set F = <*2,3,1*>, G = <*3,1,2*>;
  set f = <*1,3,2*>, g = <*2,1,3*>, h = <*3,2,1*>;
A1: dom f = {1,2,3} by FINSEQ_1:89,FINSEQ_3:1;
  then
A2: 1 in dom f by ENUMSET1:def 1;
A3: f is Permutation of Seg 3 by Th27,MATRIX_1:def 12;
A4: g is Permutation of Seg 3 by Th27,MATRIX_1:def 12;
A5: 2 in dom f by A1,ENUMSET1:def 1;
A6: dom G = {1,2,3} by FINSEQ_1:89,FINSEQ_3:1;
  then
A7: 1 in dom G by ENUMSET1:def 1;
A8: 3 in dom G by A6,ENUMSET1:def 1;
A9: 2 in dom G by A6,ENUMSET1:def 1;
A10: 3 in dom f by A1,ENUMSET1:def 1;
A11: dom g = {1,2,3} by FINSEQ_1:89,FINSEQ_3:1;
  then
A12: 1 in dom g by ENUMSET1:def 1;
A13: 3 in dom g by A11,ENUMSET1:def 1;
A14: 2 in dom g by A11,ENUMSET1:def 1;
A15: dom h = {1,2,3} by FINSEQ_1:89,FINSEQ_3:1;
  then
A16: 1 in dom h by ENUMSET1:def 1;
A17: 3 in dom h by A15,ENUMSET1:def 1;
A18: 2 in dom h by A15,ENUMSET1:def 1;
A19: f is Permutation of Seg 3 by Th27,MATRIX_1:def 12;
A20: G is Permutation of Seg 3 by Th27,MATRIX_1:def 12;
A21: dom F = {1,2,3} by FINSEQ_1:89,FINSEQ_3:1;
  then
A22: 1 in dom F by ENUMSET1:def 1;
A23: 3 in dom F by A21,ENUMSET1:def 1;
A24: 2 in dom F by A21,ENUMSET1:def 1;
A25: h is Permutation of Seg 3 by Th27,MATRIX_1:def 12;
A26: F is Permutation of Seg 3 by Th27,MATRIX_1:def 12;
  then reconsider f,g,h,F,G as FinSequence of Seg 3 by A4,A25,A20,A19,Th36;
A27: 3 = len g by FINSEQ_1:45;
  then reconsider gf = g * f as FinSequence of Seg 3 by A3,FINSEQ_2:46;
A28: gf.1 = g.(f.1) by A2,FUNCT_1:13
    .= g.1
    .= 2;
A29: g is Permutation of Seg 3 by Th27,MATRIX_1:def 12;
  then reconsider gg = g * g as FinSequence of Seg 3 by A27,FINSEQ_2:46;
A30: gg.1 = g.(g.1) by A12,FUNCT_1:13
    .= g.2
    .= 1;
A31: 3 = len f by FINSEQ_1:45;
  then reconsider fg = f * g as FinSequence of Seg 3 by A4,FINSEQ_2:46;
A32: fg.2 = f.(g.2) by A14,FUNCT_1:13
    .= f.1
    .= 1;
A33: gf.3 = g.(f.3) by A10,FUNCT_1:13
    .= g.2
    .= 1;
A34: gf.2 = g.(f.2) by A5,FUNCT_1:13
    .= g.3
    .= 3;
A35: f is Permutation of Seg 3 by Th27,MATRIX_1:def 12;
  then reconsider ff = f * f as FinSequence of Seg 3 by A31,FINSEQ_2:46;
  len gf = 3 by A27,A35,FINSEQ_2:43;
  hence <*2,1,3*> * <*1,3,2*> = <*2,3,1*> by A28,A34,A33,FINSEQ_1:45;
A36: fg.1 = f.(g.1) by A12,FUNCT_1:13
    .= f.2
    .= 3;
A37: fg.3 = f.(g.3) by A13,FUNCT_1:13
    .= f.3
    .= 2;
  len fg = 3 by A31,A29,FINSEQ_2:43;
  hence <*1,3,2*> * <*2,1,3*> = <*3,1,2*> by A36,A32,A37,FINSEQ_1:45;
A38: ff.2 = f.(f.2) by A5,FUNCT_1:13
    .= f.3
    .= 2;
A39: gg.3 = g.(g.3) by A13,FUNCT_1:13
    .= g.3
    .= 3;
A40: gg.2 = g.(g.2) by A14,FUNCT_1:13
    .= g.1
    .= 2;
A41: h is Permutation of Seg 3 by Th27,MATRIX_1:def 12;
  then reconsider gh = g * h as FinSequence of Seg 3 by A27,FINSEQ_2:46;
A42: gh.1 = g.(h.1) by A16,FUNCT_1:13
    .= g.3
    .= 3;
A43: gh.3 = g.(h.3) by A17,FUNCT_1:13
    .= g.1
    .= 2;
A44: gh.2 = g.(h.2) by A18,FUNCT_1:13
    .= g.2
    .= 1;
A45: 3 = len h by FINSEQ_1:45;
  then reconsider hf = h * f as FinSequence of Seg 3 by A19,FINSEQ_2:46;
  reconsider hh = h * h as FinSequence of Seg 3 by A45,A41,FINSEQ_2:46;
A46: hh.1 = h.(h.1) by A16,FUNCT_1:13
    .= h.3
    .= 1;
  reconsider fh = f * h as FinSequence of Seg 3 by A25,A31,FINSEQ_2:46;
A47: fh.1 = f.(h.1) by A16,FUNCT_1:13
    .= f.3
    .= 2;
A48: fh.3 = f.(h.3) by A17,FUNCT_1:13
    .= f.1
    .= 1;
  reconsider fF = f * F as FinSequence of Seg 3 by A26,A31,FINSEQ_2:46;
A49: fF.1 = f.(F.1) by A22,FUNCT_1:13
    .= f.2
    .= 3;
A50: fF.2 = f.(F.2) by A24,FUNCT_1:13
    .= f.3
    .= 2;
  reconsider hg = h * g as FinSequence of Seg 3 by A45,A29,FINSEQ_2:46;
A51: hg.1 = h.(g.1) by A12,FUNCT_1:13
    .= h.2
    .= 2;
A52: hg.2 = h.(g.2) by A14,FUNCT_1:13
    .= h.1
    .= 3;
A53: hh.3 = h.(h.3) by A17,FUNCT_1:13
    .= h.1
    .= 3;
A54: hh.2 = h.(h.2) by A18,FUNCT_1:13
    .= h.2
    .= 2;
  len gh = 3 by A27,A41,FINSEQ_2:43;
  hence <*2,1,3*> * <*3,2,1*> = <*3,1,2*> by A42,A44,A43,FINSEQ_1:45;
A55: ff.3 = f.(f.3) by A10,FUNCT_1:13
    .= f.2
    .= 3;
A56: hg.3 = h.(g.3) by A13,FUNCT_1:13
    .= h.3
    .= 1;
  len hg = 3 by A45,A29,FINSEQ_2:43;
  hence <*3,2,1*> * <*2,1,3*> = <*2,3,1*> by A51,A52,A56,FINSEQ_1:45;
  len hh = 3 by A45,A41,FINSEQ_2:43;
  hence <*3,2,1*> * <*3,2,1*> = <*1,2,3*> by A46,A54,A53,FINSEQ_1:45;
  len gg = 3 by A27,A29,FINSEQ_2:43;
  hence <*2,1,3*> * <*2,1,3*> = <*1,2,3*> by A30,A40,A39,FINSEQ_1:45;
A57: ff.1 = f.(f.1) by A2,FUNCT_1:13
    .= f.1
    .= 1;
A58: fF.3 = f.(F.3) by A23,FUNCT_1:13
    .= f.1
    .= 1;
  len ff = 3 by A31,A35,FINSEQ_2:43;
  hence <*1,3,2*> * <*1,3,2*> = <*1,2,3*> by A57,A38,A55,FINSEQ_1:45;
A59: F is Permutation of Seg 3 by Th27,MATRIX_1:def 12;
  then len fF = 3 by A31,FINSEQ_2:43;
  hence <*1,3,2*> * <*2,3,1*> = <*3,2,1*> by A49,A50,A58,FINSEQ_1:45;
A60: fh.2 = f.(h.2) by A18,FUNCT_1:13
    .= f.2
    .= 3;
A61: 3 = len F by FINSEQ_1:45;
  then reconsider FF = F * F as FinSequence of Seg 3 by A26,FINSEQ_2:46;
  reconsider FG = F * G as FinSequence of Seg 3 by A20,A61,FINSEQ_2:46;
A62: FG.1 = F.(G.1) by A7,FUNCT_1:13
    .= F.3
    .= 1;
A63: FG.2 = F.(G.2) by A9,FUNCT_1:13
    .= F.1
    .= 2;
A64: FF.3 = F.(F.3) by A23,FUNCT_1:13
    .= F.1
    .= 2;
A65: FG.3 = F.(G.3) by A8,FUNCT_1:13
    .= F.2
    .= 3;
A66: FF.2 = F.(F.2) by A24,FUNCT_1:13
    .= F.3
    .= 1;
A67: FF.1 = F.(F.1) by A22,FUNCT_1:13
    .= F.2
    .= 3;
A68: 3 = len G by FINSEQ_1:45;
  then reconsider GF = G * F as FinSequence of Seg 3 by A26,FINSEQ_2:46;
  reconsider GG = G * G as FinSequence of Seg 3 by A20,A68,FINSEQ_2:46;
A69: GG.1 = G.(G.1) by A7,FUNCT_1:13
    .= G.3
    .= 2;
A70: GG.2 = G.(G.2) by A9,FUNCT_1:13
    .= G.1
    .= 3;
A71: GF.3 = G.(F.3) by A23,FUNCT_1:13
    .= G.1
    .= 3;
A72: GG.3 = G.(G.3) by A8,FUNCT_1:13
    .= G.2
    .= 1;
A73: GF.2 = G.(F.2) by A24,FUNCT_1:13
    .= G.3
    .= 2;
A74: GF.1 = G.(F.1) by A22,FUNCT_1:13
    .= G.2
    .= 1;
  len FF = 3 by A61,A59,FINSEQ_2:43;
  hence <*2,3,1*> * <*2,3,1*> = <*3,1,2*> by A67,A66,A64,FINSEQ_1:45;
A75: G is Permutation of Seg 3 by Th27,MATRIX_1:def 12;
  then len FG = 3 by A61,FINSEQ_2:43;
  hence <*2,3,1*> * <*3,1,2*> = <*1,2,3*> by A62,A63,A65,FINSEQ_1:45;
A76: hf.3 = h.(f.3) by A10,FUNCT_1:13
    .= h.2
    .= 2;
  len GF = 3 by A68,A59,FINSEQ_2:43;
  hence <*3,1,2*> * <*2,3,1*> = <*1,2,3*> by A74,A73,A71,FINSEQ_1:45;
  len GG = 3 by A68,A75,FINSEQ_2:43;
  hence <*3,1,2*> * <*3,1,2*> = <*2,3,1*> by A69,A70,A72,FINSEQ_1:45;
A77: hf.2 = h.(f.2) by A5,FUNCT_1:13
    .= h.3
    .= 1;
  len fh = 3 by A31,A41,FINSEQ_2:43;
  hence <*1,3,2*> * <*3,2,1*> = <*2,3,1*> by A47,A60,A48,FINSEQ_1:45;
A78: hf.1 = h.(f.1) by A2,FUNCT_1:13
    .= h.1
    .= 3;
  len hf = 3 by A45,A35,FINSEQ_2:43;
  hence <*3,2,1*> * <*1,3,2*> = <*3,1,2*> by A78,A77,A76,FINSEQ_1:45;
end;
