reserve x,y for set;

theorem Th47:
  for A being transitive non empty AltCatStr for B being
  transitive non empty SubCatStr of A holds B opp is SubCatStr of A opp
proof
  let A be transitive non empty AltCatStr;
  let B be transitive non empty SubCatStr of A;
A1: B, B opp are_opposite by YELLOW18:def 4;
  then
A2: the carrier of B opp = the carrier of B by YELLOW18:def 3;
A3: the Arrows of B opp = ~the Arrows of B by A1,YELLOW18:def 3;
A4: A, A opp are_opposite by YELLOW18:def 4;
  then
A5: the carrier of A opp = the carrier of A by YELLOW18:def 3;
  hence the carrier of B opp c= the carrier of A opp by A2,ALTCAT_2:def 11;
  the Arrows of B cc= the Arrows of A & the Arrows of A opp = ~the Arrows
  of A by A4,ALTCAT_2:def 11,YELLOW18:def 3;
  hence the Arrows of B opp cc= the Arrows of A opp by A3,Th46;
A6: the carrier of B c= the carrier of A by ALTCAT_2:def 11;
  hence
  [:the carrier of B opp, the carrier of B opp, the carrier of B opp:] c=
  [:the carrier of A opp, the carrier of A opp, the carrier of A opp:] by A5,A2
,MCART_1:73;
  let x;
  assume
  x in [:the carrier of B opp, the carrier of B opp, the carrier of B opp :];
  then consider x1,x2,x3 being object such that
A7: x1 in the carrier of B & x2 in the carrier of B & x3 in the carrier
  of B and
A8: x = [x1,x2,x3] by A2,MCART_1:68;
  reconsider a = x1, b = x2, c = x3 as Object of B by A7;
  reconsider a1 = a, b1 = b, c1 = c as Object of A by A6;
  reconsider a19 = a1, b19 = b1, c19 = c1 as Object of A opp by A4,
YELLOW18:def 3;
A9: the Comp of B cc= the Comp of A & (the Comp of B).(c,b,a) = (the Comp
  of B). [c,b,a] by ALTCAT_2:def 11,MULTOP_1:def 1;
A10: (the Comp of A).(c1,b1,a1) = (the Comp of A).[c1,b1,a1] by MULTOP_1:def 1;
  [x3,x2,x1] in [:the carrier of B, the carrier of B, the carrier of B:]
  by A7,MCART_1:69;
  then
A11: (the Comp of B).(c,b,a) c= (the Comp of A).(c1,b1,a1) by A9,A10;
  reconsider a9 = a, b9 = b, c9 = c as Object of B opp by A1,YELLOW18:def 3;
A12: (the Comp of B opp).(a9,b9,c9) = (the Comp of B opp).x & (the Comp of A
  opp) .(a19,b19,c19) = (the Comp of A opp).x by A8,MULTOP_1:def 1;
A13: (the Comp of A opp).(a19,b19,c19) = ~((the Comp of A).(c1,b1,a1)) by A4,
YELLOW18:def 3;
  (the Comp of B opp).(a9,b9,c9) = ~((the Comp of B).(c,b,a)) by A1,
YELLOW18:def 3;
  hence thesis by A13,A12,A11,Th44;
end;
