theorem Th43:
  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 Def23
    .= [[dom(m2*m1),cod(m2*m1)],(m2*m1)`2] by Th41;
  hence thesis by Lm2,XTUPLE_0:1;
end;
