
theorem Th13:
  for C,C1,C2 being non empty AltCatStr st C, C1 are_opposite
  holds C, C2 are_opposite iff the AltCatStr of C1 = the AltCatStr of C2
proof
  let C, C1, C2 be non empty AltCatStr such that
A1: the carrier of C1 = the carrier of C and
A2: the Arrows of C1 = ~the Arrows of C and
A3: for a,b,c being Object of C
  for a9,b9,c9 being Object of C1 st a9 = a & b9 = b & c9 = c
  holds (the Comp of C1).(a9,b9,c9) = ~((the Comp of C).(c,b,a));
  thus
  C, C2 are_opposite implies the AltCatStr of C1 = the AltCatStr of C2
  proof
    assume that
A4: the carrier of C2 = the carrier of C and
A5: the Arrows of C2 = ~the Arrows of C and
A6: for a,b,c being Object of C
    for a9,b9,c9 being Object of C2 st a9 = a & b9 = b & c9 = c
    holds (the Comp of C2).(a9,b9,c9) = ~((the Comp of C).(c,b,a));
A7: dom the Comp of C1 = [:the carrier of C1, the carrier of C1,
    the carrier of C1:] by PARTFUN1:def 2;
A8: dom the Comp of C2 = [:the carrier of C2, the carrier of C2,
    the carrier of C2:] by PARTFUN1:def 2;
    now
      let x be object;
      assume x in [:the carrier of C, the carrier of C, the carrier of C:];
      then consider a,b,c being object such that
A9:   a in the carrier of C and
A10:  b in the carrier of C and
A11:  c in the carrier of C and
A12:  x = [a,b,c] by MCART_1:68;
      reconsider a, b, c as Object of C by A9,A10,A11;
      reconsider a1 = a, b1 = b, c1 = c as Object of C1 by A1;
      reconsider a2 = a, b2 = b, c2 = c as Object of C2 by A4;
A13:  (the Comp of C1).(a1,b1,c1) = ~((the Comp of C).(c,b,a)) by A3;
      (the Comp of C2).(a2,b2,c2) = ~((the Comp of C).(c,b,a)) by A6;
      hence (the Comp of C1).x = (the Comp of C2).(a2,b2,c2)
      by A12,A13,MULTOP_1:def 1
        .= (the Comp of C2).x by A12,MULTOP_1:def 1;
    end;
    hence thesis by A1,A2,A4,A5,A7,A8,FUNCT_1:2;
  end;
  assume
A14: the AltCatStr of C1 = the AltCatStr of C2;
  hence the carrier of C2 = the carrier of C &
  the Arrows of C2 = ~the Arrows of C by A1,A2;
  let a,b,c be Object of C, a9,b9,c9 be Object of C2;
  thus thesis by A3,A14;
end;
