let A be non empty transitive AltCatStr ; for B being non empty transitive SubCatStr of A holds B opp is SubCatStr of A opp
let B be non empty transitive SubCatStr of A; B opp is SubCatStr of A opp
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; ALTCAT_2:def 11 ( the Arrows of (B opp) cc= the Arrows of (A opp) & the Comp of (B opp) cc= the Comp of (A opp) )
( 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; the Comp of (B opp) cc= the Comp of (A opp)
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; ALTCAT_2:def 2 for b1 being set holds
( not b1 in [: the carrier of (B opp), the carrier of (B opp), the carrier of (B opp):] or the Comp of (B opp) . b1 c= the Comp of (A opp) . b1 )
let x be set ; ( not x in [: the carrier of (B opp), the carrier of (B opp), the carrier of (B opp):] or the Comp of (B opp) . x c= the Comp of (A opp) . x )
assume
x in [: the carrier of (B opp), the carrier of (B opp), the carrier of (B opp):]
; the Comp of (B opp) . x c= the Comp of (A opp) . x
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
the Comp of (B opp) . x c= the Comp of (A opp) . x
by A13, A12, A11, Th44; verum