reserve x,y,z for Element of REAL+;

theorem
  y <=' x & not y <=' z implies x - (y -' z) = x -' y + z
proof
  assume that
A1: y <=' x and
A2: not y <=' z;
  y -' z <=' y by Th11;
  then y -' z <=' x by A1,Th3;
  then x - (y -' z) = x -' (y -' z) by Def2;
  hence thesis by A1,A2,Lm12;
end;
