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 in dom h implies dom(h*(A --> x)) <> {}
proof
  assume that
A1: A <> {} and
A2: x in dom h;
  set y = the Element of A;
A3: y in dom (A -->x) by A1;
  (A --> x).y = x by A1,Th7;
  hence thesis by A2,A3,FUNCT_1:11;
end;
