reserve A,B,X,X1,Y,Y1,Y2,Z for set, a,x,y,z for object;
reserve P,R for Relation of X,Y;
reserve D,D1,D2,E,F for non empty set;
reserve R for Relation of D,E;
reserve x for Element of D;
reserve y for Element of E;

theorem
  for P being Relation of D,E, R being Relation of E,F
   for x, z being object holds [x,z] in P*R iff
    ex y being Element of E st [x,y] in P & [y,z] in R
proof
  let P be Relation of D,E, R be Relation of E,F;
  let x, z be object;
  thus [x,z] in P*R implies ex y being Element of E st [x,y] in P & [y,z] in R
  proof
    assume [x,z] in P*R;
    then consider y being object such that
A1: [x,y] in P and
A2: [y,z] in R by RELAT_1:def 8;
    reconsider a = y as Element of E by A1,ZFMISC_1:87;
    take a;
    thus thesis by A1,A2;
  end;
  given y such that
A3: [x,y] in P & [y,z] in R;
  thus thesis by A3,RELAT_1:def 8;
end;
