reserve X for set,
        A for Subset of X,
        R,S for Relation of X;

theorem Th3:
  R * S = {[x,y] where x,y is Element of X : ex z being Element of X st
  [x,z] in R & [z,y] in S}
  proof
    reconsider RS = R * S as Relation of X;
    A1: R * S c= {[x,y] where x,y is Element of X : ex z being Element of X
    st [x,z] in R & [z,y] in S}
    proof
      let t be object;
      assume
A1:   t in R * S;
      consider x1,x2 be object such that
      x1 in X and
A2:   x2 in X and
A3:   t = [x1,x2] by A1,ZFMISC_1:def 2;
      consider z be object such that
A4:   [x1,z] in R and
A5:   [z,x2] in S by A1,A3,RELAT_1:def 8;
      reconsider x1,x2,z1 = z as Element of X by A2,A4,ZFMISC_1:87;
      [x1,z1] in R & [z1,x2] in S by A4,A5;
      hence thesis by A3;
    end;
    {[x,y] where x,y is Element of X : ex z being Element of X st [x,z] in R &
    [z,y] in S} c= R * S
    proof
      let t be object;
      assume t in {[x,y] where x,y is Element of X : ex z being Element of X
      st [x,z] in R & [z,y] in S};
      then ex x,y be Element of X st t = [x,y] & ex z being Element of X st
      [x,z] in R & [z,y] in S;
      hence thesis by RELAT_1:def 8;
    end;
    hence thesis by A1;
  end;
