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
  dom f c= X implies SignGen(f,B,X) = (the_inverseOp_wrt B)*f
proof
  assume
A1: dom f c= X;
  rng f c= D=dom (the_inverseOp_wrt B) by FUNCT_2:def 1;
  then
A2: dom ((the_inverseOp_wrt B)*f) = dom f = dom SignGen(f,B,X)
    by Def11,RELAT_1:27;
  for k st k in dom SignGen(f,B,X) holds SignGen(f,B,X).k=
    ((the_inverseOp_wrt B)*f).k
  proof
    let k such that
A3:   k in dom SignGen(f,B,X);
    SignGen(f,B,X).k = (the_inverseOp_wrt B).(f.k) by A1,A3,A2,Def11;
    hence thesis by A3,A2,FUNCT_1:12;
  end;
  hence thesis by A2;
end;
