reserve A,X,X1,X2,Y,Y1,Y2 for set, a,b,c,d,x,y,z for object;
reserve P,P1,P2,Q,R,S for Relation;

theorem Th30:
  (P*R)*Q = P*(R*Q)
proof
  let a,b;
  hereby
    assume [a,b] in (P*R)*Q;
    then consider y such that
A1: [a,y] in P*R and
A2: [y,b] in Q by Def6;
    consider x such that
A3: [a,x] in P and
A4: [x,y] in R by A1,Def6;
    [x,b] in R*Q by A2,A4,Def6;
    hence [a,b] in P*(R*Q) by A3,Def6;
  end;
  assume [a,b] in P*(R*Q);
  then consider y such that
A5: [a,y] in P and
A6: [y,b] in R*Q by Def6;
    consider x such that
A7: [y,x] in R and
A8: [x,b] in Q by A6,Def6;
    [a,x] in P*R by A5,A7,Def6;
    hence [a,b] in (P*R)*Q by A8,Def6;
  end;
