
theorem Th14:
  for k,x1,x2,y1,y2 being Nat st x2 <= k & x1 <= (k-'x2)+y2
  holds x2+(x1-'y2) <= k &
  (((k-'x2)+y2)-'x1)+y1 = (k-'(x2+(x1-'y2)))+((y2-'x1)+y1)
  proof
    let k,x1,x2,y1,y2 be Nat;
    assume Z0: x2 <= k;
    then A1: k-'x2 = k-x2 by XREAL_1:233;
    assume Z1: x1 <= (k-'x2)+y2;
    thus x2+(x1-'y2) <= k
    proof
      per cases;
      suppose x1 < y2;
        then x1-'y2 = 0 by NAT_2:8;
        hence thesis by Z0;
      end;
      suppose x1 >= y2;
        then x1-'y2 = x1-y2 by XREAL_1:233;
        then x1-'y2 <= (k-x2)+y2-y2 = k-x2 by A1,Z1,XREAL_1:9;
        then x2+(x1-'y2) <= k-x2+x2 by XREAL_1:6;
        hence thesis;
      end;
    end;
    then
A2: k-'(x2+(x1-'y2)) = k-(x2+(x1-'y2)) & ((k-'x2)+y2)-'x1 = ((k-x2)+y2)-x1
    by Z1,A1,XREAL_1:233;
    per cases;
    suppose x1 <= y2;
      then x1-'y2 = 0 & y2-'x1 = y2-x1 by XREAL_1:233,NAT_2:8;
      hence (((k-'x2)+y2)-'x1)+y1 = (k-'(x2+(x1-'y2)))+((y2-'x1)+y1) by A2;
    end;
    suppose x1 > y2;
      then y2-'x1 = 0 & x1-'y2 = x1-y2 by XREAL_1:233,NAT_2:8;
      hence (((k-'x2)+y2)-'x1)+y1 = (k-'(x2+(x1-'y2)))+((y2-'x1)+y1) by A2;
    end;
  end;
