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 Th89:
  SignGen(f,B,X) = SignGen(f,B,X/\dom f)
proof
A1: dom SignGen(f,B,X)= dom f =dom SignGen(f,B,X/\dom f) by Def11;
  for i st i in dom SignGen(f,B,X) holds
    SignGen(f,B,X).i = SignGen(f,B,X/\dom f).i
  proof
    let i such that
A2:   i in dom SignGen(f,B,X);
    per cases;
    suppose i in X;
      then SignGen(f,B,X).i = (the_inverseOp_wrt B).(f.i) & i in X/\dom f
        by Def11,A1,XBOOLE_0:def 4,A2;
      hence thesis by Def11,A1,A2;
    end;
    suppose not i in X;
      then SignGen(f,B,X).i = f.i & not i in X/\dom f
        by Def11,XBOOLE_0:def 4,A2;
      hence thesis by Def11,A1,A2;
    end;
  end;
  hence thesis by A1;
end;
