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;
