reserve V for non empty set,
  A,B,A9,B9 for Element of V;
reserve f,f9 for Element of Funcs(V);
reserve m,m1,m2,m3,m9 for Element of Maps V;

theorem Th12:
  dom m2 = cod m1 implies (m2*m1)`2 = m2`2*m1`2 & dom(m2*m1) = dom
  m1 & cod(m2*m1) = cod m2
proof
  assume dom m2 = cod m1;
  then [[dom m1,cod m2],m2`2*m1`2] = m2*m1 by Def6
    .= [[dom(m2*m1),cod(m2*m1)],(m2*m1)`2] by Th8;
  hence thesis by Lm1,XTUPLE_0:1;
end;
