reserve X for set, R,R1,R2 for Relation;
reserve x,y,z for set;
reserve n,m,k for Nat;

theorem
  (R\,R1)\,R2 = R\,(R1\,R2)
  proof
    let x,y be object;
    reconsider xx = x, yy = y as set by TARSKI:1;
    thus [x,y] in (R\,R1)\,R2 implies [x,y] in R\,(R1\,R2)
    proof
      assume [x,y] in (R\,R1)\,R2; then
      xx,yy in (R\,R1)\,R2; then
      xx,yy in R\,R1 or yy,xx nin R\,R1 & xx,yy in R2 by Th9; then
      xx,yy in R or yy,xx nin R & xx,yy in R1 or
      yy,xx nin R & (xx,yy in R or yy,xx nin R1) & xx,yy in R2 by Th9; then
      xx,yy in R or yy,xx nin R & xx,yy in R1\,R2 by Th9; then
      xx,yy in R\,(R1\,R2) by Th9;
      hence thesis;
    end;
    assume [x,y] in R\,(R1\,R2); then
    xx,yy in R\,(R1\,R2); then
    xx,yy in R or yy,xx nin R & xx,yy in R1\,R2 by Th9; then
    xx,yy in R or yy,xx nin R & (xx,yy in R1 or yy,xx nin R1 &
      xx,yy in R2) by Th9;then
    xx,yy in R\,R1 or yy,xx nin R\,R1 & xx,yy in R2 by Th9; then
    xx,yy in (R\,R1)\,R2 by Th9;
    hence thesis;
  end;
