reserve a,b for object, I,J for set;

theorem Th15:
  for k,x1,x2,y1,y2 being bag of I st x2 divides k & x1 divides (k-'x2)+y2
  holds x2+(x1-'y2) divides k &
  (((k-'x2)+y2)-'x1)+y1 = (k-'(x2+(x1-'y2)))+((y2-'x1)+y1)
  proof let k,x1,x2,y1,y2 be bag of I;
    assume
A1: for a holds x2.a <= k.a;
    assume
A2: for a holds x1.a <= ((k-'x2)+y2).a;
    hereby
      let a;
      x2.a <= k.a & x1.a <= ((k-'x2)+y2).a = (k-'x2).a+y2.a = (k.a-'x2.a)+y2.a
      by A1,A2,PRE_POLY:def 5,def 6;
      then x2.a+(x1.a-'y2.a) <= k.a by Th14;
      then x2.a+(x1-'y2).a <= k.a by PRE_POLY:def 6;
      hence (x2+(x1-'y2)).a <= k.a by PRE_POLY:def 5;
    end;
    let a such that a in I;
    x2.a <= k.a & x1.a <= ((k-'x2)+y2).a = (k-'x2).a+y2.a = (k.a-'x2.a)+y2.a
    by A1,A2,PRE_POLY:def 5,def 6;
    then
A3: (((k.a-'x2.a)+y2.a)-'x1.a)+y1.a =
    (k.a-'(x2.a+(x1.a-'y2.a)))+((y2.a-'x1.a)+y1.a) by Th14;
A4: (((k.a-'x2.a)+y2.a)-'x1.a)+y1.a
    = (((k-'x2).a+y2.a)-'x1.a)+y1.a by PRE_POLY:def 6
    .= (((k-'x2)+y2).a-'x1.a)+y1.a by PRE_POLY:def 5
    .= (((k-'x2)+y2)-'x1).a+y1.a by PRE_POLY:def 6
    .= ((((k-'x2)+y2)-'x1)+y1).a by PRE_POLY:def 5;
    (k.a-'(x2.a+(x1.a-'y2.a)))+((y2.a-'x1.a)+y1.a)
    = (k.a-'(x2.a+(x1.a-'y2.a)))+((y2-'x1).a+y1.a) by PRE_POLY:def 6
    .= (k.a-'(x2.a+(x1-'y2).a))+((y2-'x1).a+y1.a) by PRE_POLY:def 6
    .= (k.a-'(x2+(x1-'y2)).a)+((y2-'x1).a+y1.a) by PRE_POLY:def 5
    .= (k.a-'(x2+(x1-'y2)).a)+((y2-'x1)+y1).a by PRE_POLY:def 5
    .= (k-'(x2+(x1-'y2))).a+((y2-'x1)+y1).a by PRE_POLY:def 6;
    hence thesis by A3,A4,PRE_POLY:def 5;
  end;
