theorem
  d in f.:X implies ex c st c in dom f & c in X & d = f.c
proof
  assume d in f.:X;
  then consider x being object such that
A1: x in dom f and
A2: x in X & d = (f qua Function).x by FUNCT_1:def 6;
  reconsider x as Element of C by A1;
  take x;
  thus thesis by A1,A2;
end;
