reserve i,j,n,k,m for Nat,
     a,b,x,y,z for object,
     F,G for FinSequence-yielding FinSequence,
     f,g,p,q for FinSequence,
     X,Y for set,
     D for non empty set;
reserve
  B,A,M for BinOp of D,
  F,G for D* -valued FinSequence,
  f for FinSequence of D,
  d,d1,d2 for Element of D;
reserve
  F,G for non-empty non empty FinSequence of D*,
  f for non empty FinSequence of D;
reserve f,g for FinSequence of D,
        a,b,c for set,
        F,F1,F2 for finite set;

theorem Th81:
  for E1 be Enumeration of F1,E2 be Enumeration of F2,
      E12 be Enumeration of F1\/F2 st E12=E1^E2 holds
  SignGenOp(f,B,F1\/F2) * E12 =
    (SignGenOp(f,B,F1) * E1)^(SignGenOp(f,B,F2) * E2)
proof
  set C1=SignGenOp(f,B,F1),C2=SignGenOp(f,B,F2),C12=SignGenOp(f,B,F1\/F2);
  let E1 be Enumeration of F1,E2 be Enumeration of F2,
      E12 be Enumeration of F1\/F2 such that
A1: E12=E1^E2;
A2: len (C1*E1)= len E1 = card F1 & len (C2*E2)= len E2 = card F2 &
    len (C12*E12)= len E12 = card (F1\/F2) by CARD_1:def 7;
  F1=rng E1 misses rng E2=F2 by A1,FINSEQ_3:91,RLAFFIN3:def 1;
  then
A3: card (F1\/F2)=card F1 + card F2 by CARD_2:40;
A4: len (C12*E12) = len ((C1*E1)^(C2*E2)) by A2,A3,FINSEQ_1:22;
A5: dom E1 = dom (C1*E1) & dom E2 = dom (C2*E2) by A2,FINSEQ_3:30;
  for k st 1 <= k & k <= len (C12*E12) holds (C12*E12).k = ((C1*E1)^(C2*E2)).k
  proof
    let k such that
A6:   1 <= k & k <= len (C12*E12);
A7:   k in dom (C12*E12) by A6,FINSEQ_3:25;
    k in dom ((C1*E1)^(C2*E2)) by A6,A4,FINSEQ_3:25;
    then per cases by FINSEQ_1:25;
    suppose
A8:     k in dom (C1*E1);
      hence ((C1*E1)^(C2*E2)).k = (C1*E1).k by FINSEQ_1:def 7
        .= SignGen(f,B,E1.k) by A8,Th80
        .= SignGen(f,B,E12.k) by A1,A8,A5,FINSEQ_1:def 7
        .= (C12*E12).k by A7,Th80;
    end;
    suppose ex i st i in dom (C2*E2) & k = len (C1*E1)+i;
      then consider i such that
A9:     i in dom (C2*E2) & k = len (C1*E1)+i;
      thus ((C1*E1)^(C2*E2)).k = (C2*E2).i by A9,FINSEQ_1:def 7
        .= SignGen(f,B,E2.i) by A9,Th80
        .= SignGen(f,B,E12.k) by A1,A9,A5,A2,FINSEQ_1:def 7
        .= (C12*E12).k by A7,Th80;
    end;
  end;
  hence thesis by A2,A3,FINSEQ_1:22;
end;
