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