reserve x,y,X,Y for set;
reserve C,D,E for non empty set;
reserve SC for Subset of C;
reserve SD for Subset of D;
reserve SE for Subset of E;
reserve c,c1,c2 for Element of C;
reserve d,d1,d2 for Element of D;
reserve e for Element of E;
reserve f,f1,g for PartFunc of C,D;
reserve t for PartFunc of D,C;
reserve s for PartFunc of D,E;
reserve h for PartFunc of C,E;
reserve F for PartFunc of D,D;

theorem
  (for c st c in dom f holds f/.c = d) implies f = dom f --> d
proof
  assume
A1: for c st c in dom f holds f/.c = d;
  now
    let x be object;
    assume
A2: x in dom f;
    then reconsider x1=x as Element of C;
    f/.x1 = d by A1,A2;
    hence ( f qua Function).x = d by A2,PARTFUN1:def 6;
  end;
  hence thesis by FUNCOP_1:11;
end;
