
theorem Th6: :: stolen from WAYBEL_1:8
  for R, S, T being RelStr for f being Function of R, S st f is
isomorphic for g being Function of S, T st g is isomorphic for h being Function
  of R, T st h = g*f holds h is isomorphic
proof
  let L1,L2,L3 be RelStr;
  let f1 be Function of L1,L2 such that
A1: f1 is isomorphic;
  let f2 be Function of L2,L3 such that
A2: f2 is isomorphic;
  let h be Function of L1,L3 such that
A3: h = f2*f1;
  per cases;
  suppose
    L1 is non empty & L2 is non empty & L3 is non empty;
    then reconsider L1,L2,L3 as non empty RelStr;
    reconsider f1 as Function of L1,L2;
    reconsider f2 as Function of L2,L3;
    consider g1 being Function of L2,L1 such that
A4: g1 = (f1 qua Function)" & g1 is monotone by A1,WAYBEL_0:def 38;
    consider g2 being Function of L3,L2 such that
A5: g2 = (f2 qua Function)" & g2 is monotone by A2,WAYBEL_0:def 38;
A6: f2*f1 is monotone by A1,A2,YELLOW_2:12;
    g1*g2 is monotone & g1*g2 = ((f2*f1) qua Function)" by A1,A2,A4,A5,
FUNCT_1:44,YELLOW_2:12;
    hence thesis by A1,A2,A3,A6,WAYBEL_0:def 38;
  end;
  suppose
A7: L1 is empty;
    then L2 is empty by A1,WAYBEL_0:def 38;
    then L3 is empty by A2,WAYBEL_0:def 38;
    hence thesis by A7,WAYBEL_0:def 38;
  end;
  suppose
    L2 is empty;
    then L1 is empty & L3 is empty by A1,A2,WAYBEL_0:def 38;
    hence thesis by WAYBEL_0:def 38;
  end;
  suppose
A8: L3 is empty;
    then L2 is empty by A2,WAYBEL_0:def 38;
    then L1 is empty by A1,WAYBEL_0:def 38;
    hence thesis by A8,WAYBEL_0:def 38;
  end;
end;
