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
  (P*R)~ = R~*P~
proof
  let a,b;
  hereby
    assume [a,b] in (P*R)~;
    then [b,a] in P*R by Def5;
    then consider y such that
A1: [b,y] in P & [y,a] in R by Def6;
    [y,b] in P~ & [a,y] in R~ by A1,Def5;
    hence [a,b] in R~*P~ by Def6;
  end;
  assume [a,b] in R~*P~;
  then consider y such that
A2: [a,y] in R~ & [y,b] in P~ by Def6;
  [y,a] in R & [b,y] in P by A2,Def5;
  then [b,a] in P*R by Def6;
  hence [a,b] in (P*R)~ by Def5;
end;
