reserve c,c1,c2,x,y,z,z1,z2 for set;
reserve C1,C2,C3 for non empty set;

theorem
  for f,g be RMembership_Func of C1,C2, h,k be RMembership_Func of C2,C3
  st f c= g & h c= k holds f(#)h c= g(#)k
proof
  let f,g be RMembership_Func of C1,C2, h,k being RMembership_Func of C2,C3;
  assume that
A1: f c= g and
A2: h c= k;
  let c be Element of [:C1,C3:];
  consider x,z being object such that
A3: x in C1 and
A4: z in C3 and
A5: c = [x,z] by ZFMISC_1:def 2;
  for y be set st y in C2 holds f. [x,y]<=g. [x,y] & h. [y,z]<=k. [y,z]
  proof
    let y be set;
    assume
A6: y in C2;
    then
A7: [y,z] in [:C2,C3:] by A4,ZFMISC_1:87;
    [x,y] in [:C1,C2:] by A3,A6,ZFMISC_1:87;
    hence thesis by A1,A2,A7;
  end;
  hence thesis by A3,A4,A5,Th18;
end;
