reserve x,y for set,
  G for Group,
  A,B,H,H1,H2 for Subgroup of G,
  a,b,c for Element of G,
  F,F1 for FinSequence of the carrier of G,
  I,I1 for FinSequence of INT,
  i,j for Element of NAT;

theorem Th5:
  [.a,b",c.] |^ b = [.[.a,b".]|^ b,c|^ b.]
proof
A1:[.a,b",c.] |^ b
    = b" * ([.a,b".]" * 1_G * c" * [.a,b".] * c) * b by GROUP_1:def 4
   .= b" * (([.a,b".]" * (b * b")) * c" * [.a,b".] * c) * b by GROUP_1:def 5
   .= b" * (([.a,b".]" * b * b") * c" * [.a,b".] * c) * b by GROUP_1:def 3
   .= b" * ((([.a,b".]" * b) * b") * c" * ([.a,b".] * c)) * b by GROUP_1:def 3
   .= b" * (([.a,b".]" * b) * b" * (c" * ([.a,b".] * c))) * b by GROUP_1:def 3
   .= (b" * (([.a,b".]" * b) * (b" * (c" * ([.a,b".] * c))))) * b
       by GROUP_1:def 3
   .= ((b" * ([.a,b".]" * b)) * (b" * (c" * ([.a,b".] * c)))) * b
       by GROUP_1:def 3
   .= (([.a,b".]"|^ b) * (b" * (c" * ([.a,b".] * c)))) * b by GROUP_1:def 3
   .= [.a,b".]"|^ b * ((b" * (c" * ([.a,b".] * c))) * b) by GROUP_1:def 3
   .= [.a,b".]"|^ b * ([.a,b".]|^ c |^ b) by GROUP_1:def 3
   .= [.b",a.]|^ b * ([.a,b".]|^ c |^ b) by GROUP_5:22
   .= [.b",a.]|^ b * [.a,b".]|^ (c * b) by GROUP_3:24;
  [.[.a,b".]|^ b,c|^ b.]
    = [.b",a.]|^ b * (c|^ b)" * ([.a,b".]|^ b) * (c|^ b) by Th4
   .= [.b",a.]|^ b * (c"|^ b) * ([.a,b".]|^ b) * (c|^ b) by GROUP_3:26
   .= [.b",a.]|^ b * (b" * c" * b) * ((b" * [.a,b".]) * b) * (b" * (c * b))
       by GROUP_1:def 3
   .= [.b",a.]|^ b * (b" * c" * b) * ((b" * [.a,b".]) * b * (b" * (c * b)))
       by GROUP_1:def 3
   .= [.b",a.]|^ b * ((b" * c") * b) * ((b" * [.a,b".]) * (b * (b" * (c * b))))
       by GROUP_1:def 3
   .= [.b",a.]|^ b * ((b" * c") * b) * ((b" * [.a,b".]) * (b * b" * (c * b)))
       by GROUP_1:def 3
   .= [.b",a.]|^ b * ((b" * c") * b) * ((b" * [.a,b".]) * (1_G * (c * b)))
       by GROUP_1:def 5
   .= [.b",a.]|^ b * ((b" * c") * b) * ((b" * [.a,b".]) * (c * b))
       by GROUP_1:def 4
   .= [.b",a.]|^ b * (((b" * c") * b) * ((b" * [.a,b".]) * (c * b)))
       by GROUP_1:def 3
   .= [.b",a.]|^ b * ((b" * c") * (b * ((b" * [.a,b".]) * (c * b))))
       by GROUP_1:def 3
   .= [.b",a.]|^ b * (((b" * c") * ((b * (b" * [.a,b".])) * (c * b))))
       by GROUP_1:def 3
   .= [.b",a.]|^ b * (((b" * c") * ((b * b" * [.a,b".]) * (c * b))))
       by GROUP_1:def 3
   .= [.b",a.]|^ b * (((b" * c") * ((1_G * [.a,b".]) * (c * b))))
       by GROUP_1:def 5
   .= [.b",a.]|^ b * (((b" * c") * ([.a,b".] * (c * b)))) by GROUP_1:def 4
   .= [.b",a.]|^ b * (((b" * c") * [.a,b".] * (c * b))) by GROUP_1:def 3
   .= [.b",a.]|^ b * [.a,b".] |^ (c * b) by GROUP_1:17;
  hence thesis by A1;
end;
