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 Th80:
for E be Enumeration of F st i in dom E or i in dom (SignGenOp(f,B,F)*E)
  holds (SignGenOp(f,B,F) * E).i = SignGen(f,B,E.i)
proof
  let E be Enumeration of F such that
A1: i in dom E or i in dom (SignGenOp(f,B,F)*E);
  set C=SignGenOp(f,B,F);
  i in dom (C*E)
  proof
    assume
A2:   not i in dom (C*E);
    then E.i in rng E = F =dom C
      by A1,FUNCT_2:def 1,FUNCT_1:def 3,RLAFFIN3:def 1;
    hence thesis by A2,A1,FUNCT_1:11;
  end;
  then (C*E).i = C.(E.i) & E.i in dom C = F by FUNCT_1:11,12,FUNCT_2:def 1;
  hence thesis by Def12;
end;
