reserve U for Universe;
reserve x for Element of U;
reserve U1,U2 for Universe;

theorem Th60:
  for o1,o2,m1,m2 being object holds 1Cat(o1,m1) ~= 1Cat(o2,m2)
  proof
    let o1,o2,m1,m2 be object;
    set C = 1Cat(o1,m1),
        C9= 1Cat(o2,m2);
A1: C = CatStr(# { o1 },{ m1 },  m1 :-> o1 , m1 :-> o1,(m1,m1):->m1 #)
      by CAT_1:def 11;
A2: C9 = CatStr(# { o2 },{ m2 },  m2 :-> o2 , m2 :-> o2,(m2,m2):->m2 #)
      by CAT_1:def 11;
    reconsider o19 =o1 as Element of C by A1,TARSKI:def 1;
    reconsider o29=o2 as Element of C9 by A2,TARSKI:def 1;
    reconsider m19=m1 as Element of the carrier' of C by A1,TARSKI:def 1;
    reconsider m29=m2 as Element of the carrier' of C9 by A2,TARSKI:def 1;
    set F = m1 :-> m2;
    reconsider F as Function of the carrier' of C,the carrier' of C9 by A1,A2;
    now
      hereby
        let c be Element of C;
A3:     F.(id c) = F.m1 by Th59 .= m2 by FUNCOP_1:72;
        reconsider c9 = o2 as Element of C9 by TARSKI:def 1,A2;
        id c9 = m2 by Th59;
        hence ex d being Element of C9 st F.(id c) = id d by A3;
      end;
      hereby
        let f be Element of the carrier' of C;
        thus F.(id dom f) = F.m1 by Th59
                         .= m2 by FUNCOP_1:72
                         .= id dom (F.f) by Th59;
        thus F.(id(cod f)) = F.m1 by Th59
                          .= m2 by FUNCOP_1:72
                          .= id (cod (F.f)) by Th59;
      end;
      thus for f,g be Element of the carrier' of C st
        [g,f] in dom the Comp of C holds F.(g(*)f) = (F.g)(*)(F.f)
        by A1,A2,TARSKI:def 1;
    end;
    then reconsider F as Functor of C,C9 by CAT_1:def 21;
    now
      thus F is one-to-one;
      thus rng F = the carrier' of C9 by A2;
A4:   for c be Element of C ex c9 be Element of C9 st F.(id c) = id c9
      proof
        let c be Element of C;
A5:     F.(id c) = F.m1 by Th59
                .= m2 by FUNCOP_1:72;
        reconsider c9 = o2 as Element of C9 by A2,TARSKI:def 1;
        id c9 = m2 by Th59;
        hence thesis by A5;
      end;
      reconsider o19 = o1 as Element of C by A1,TARSKI:def 1;
      reconsider o29 = o2 as Element of C9 by A2,TARSKI:def 1;
A6:   F.(id o19) = F.m1 by Th59
                .= m2 by FUNCOP_1:72
                .= id o29 by Th59;
      dom Obj F = {o1} by A1,PARTFUN1:def 2;
      then rng Obj F = {(Obj F).o19} by FUNCT_1:4
                    .= {o29} by A6,A4,CAT_1:def 22;
      hence rng Obj F = the carrier of C9 by A2;
    end;
    hence thesis by CAT_1:def 25,ISOCAT_1:def 4;
  end;
