reserve f,g,h for Function,
  A for set;
reserve F for Function,
  B,x,y,y1,y2,z for set;
reserve x,z for object;

theorem
  x in dom h implies h*(A --> x) = A --> h.x
proof
  assume
A1: x in dom h;
A2: now
    let z be object;
    assume
A3: z in dom (h*(A --> x));
    then z in dom (A --> x) by FUNCT_1:11;
    then
A4: z in A;
    thus (h*(A --> x)).z = h.((A --> x).z) by A3,FUNCT_1:12
      .= h.x by A4,Th7
      .= (A --> h.x).z by A4,Th7;
  end;
  dom (h*(A --> x)) = (A --> x)"dom h by RELAT_1:147
    .= A by A1,Th14
    .= dom (A --> h.x);
  hence thesis by A2;
end;
