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 Th29:
  F[:](f,x)*h = F[:](f*h,x)
proof
  dom (dom f -->x) = dom f; then
A2: dom ((dom f --> x)*h) = dom (f*h) by RELAT_1:163;
A3: now
    let z;
    assume
A4: z in dom ((dom f --> x)*h);
    then
A5: h.z in dom(dom f -->x) by FUNCT_1:11;
    thus ((dom f --> x)*h).z = (dom f --> x).(h.z) by A4,FUNCT_1:12
      .= x by A5,Th7;
  end;
  thus F[:](f,x)*h = F.:(f, dom f --> x)*h
    .= F.:(f*h, (dom f --> x)*h) by Th25
    .= F[:](f*h,x) by A2,A3,Th11;
end;
