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
  for h st dom h = dom(F.:(f,g)) & for z being set st z in dom (F.:(f,g)
  ) holds h.z = F.(f.z,g.z) holds h = F.:(f,g)
proof
  let h;
  assume that
A1: dom h = dom(F.:(f,g)) and
A2: for z being set st z in dom (F.:(f,g)) holds h.z = F.(f.z,g.z);
  now
    let z be object;
    assume
A3: z in dom (F.:(f,g));
    hence h.z = F.(f.z,g.z) by A2
      .= (F.:(f,g)).z by A3,Lm1;
  end;
  hence thesis by A1;
end;
