deffunc H1( Element of H) -> Function of (the_sylow_p-subgroups_of_prime (p,G)),(the_sylow_p-subgroups_of_prime (p,G)) = the_left_translation_of ($1,p);
set E = the_sylow_p-subgroups_of_prime (p,G);
A1: for h1, h2 being Element of H holds H1(h1 * h2) = H1(h1) * H1(h2)
proof
let h1, h2 be Element of H; :: thesis: H1(h1 * h2) = H1(h1) * H1(h2)
set f12 = the_left_translation_of ((h1 * h2),p);
set f1 = the_left_translation_of (h1,p);
set f2 = the_left_translation_of (h2,p);
the_left_translation_of (h1,p) in Funcs ((the_sylow_p-subgroups_of_prime (p,G)),(the_sylow_p-subgroups_of_prime (p,G))) by FUNCT_2:9;
then A2: ex f being Function st
( the_left_translation_of (h1,p) = f & dom f = the_sylow_p-subgroups_of_prime (p,G) & rng f c= the_sylow_p-subgroups_of_prime (p,G) ) by FUNCT_2:def 2;
the_left_translation_of (h2,p) in Funcs ((the_sylow_p-subgroups_of_prime (p,G)),(the_sylow_p-subgroups_of_prime (p,G))) by FUNCT_2:9;
then A3: ex f being Function st
( the_left_translation_of (h2,p) = f & dom f = the_sylow_p-subgroups_of_prime (p,G) & rng f c= the_sylow_p-subgroups_of_prime (p,G) ) by FUNCT_2:def 2;
A4: now :: thesis: for x being object st x in dom (the_left_translation_of ((h1 * h2),p)) holds
(the_left_translation_of ((h1 * h2),p)) . x = (the_left_translation_of (h1,p)) . ((the_left_translation_of (h2,p)) . x)
let x be object ; :: thesis: ( x in dom (the_left_translation_of ((h1 * h2),p)) implies (the_left_translation_of ((h1 * h2),p)) . x = (the_left_translation_of (h1,p)) . ((the_left_translation_of (h2,p)) . x) )
assume A5: x in dom (the_left_translation_of ((h1 * h2),p)) ; :: thesis: (the_left_translation_of ((h1 * h2),p)) . x = (the_left_translation_of (h1,p)) . ((the_left_translation_of (h2,p)) . x)
then reconsider P19 = x as Element of the_sylow_p-subgroups_of_prime (p,G) ;
reconsider P1999 = x as Element of the_sylow_p-subgroups_of_prime (p,G) by A5;
consider P29 being Element of the_sylow_p-subgroups_of_prime (p,G), H19, H29 being strict Subgroup of G, g2 being Element of G such that
A6: ( P29 = (the_left_translation_of (h2,p)) . P19 & P19 = H19 & P29 = H29 ) and
A7: h2 " = g2 and
A8: H29 = H19 |^ g2 by Def20;
(the_left_translation_of (h2,p)) . x in rng (the_left_translation_of (h2,p)) by A3, A5, FUNCT_1:3;
then reconsider P199 = (the_left_translation_of (h2,p)) . x as Element of the_sylow_p-subgroups_of_prime (p,G) ;
consider P299 being Element of the_sylow_p-subgroups_of_prime (p,G), H199, H299 being strict Subgroup of G, g1 being Element of G such that
A9: P299 = (the_left_translation_of (h1,p)) . P199 and
A10: ( P199 = H199 & P299 = H299 ) and
A11: h1 " = g1 and
A12: H299 = H199 |^ g1 by Def20;
consider P2999 being Element of the_sylow_p-subgroups_of_prime (p,G), H1999, H2999 being strict Subgroup of G, g3 being Element of G such that
A13: P2999 = (the_left_translation_of ((h1 * h2),p)) . P1999 and
A14: P1999 = H1999 and
A15: P2999 = H2999 and
A16: (h1 * h2) " = g3 and
A17: H2999 = H1999 |^ g3 by Def20;
g3 = (h2 ") * (h1 ") by A16, GROUP_1:17;
then P2999 = H1999 |^ (g2 * g1) by A7, A11, A15, A17, GROUP_2:43
.= P299 by A6, A8, A10, A12, A14, GROUP_3:60 ;
hence (the_left_translation_of ((h1 * h2),p)) . x = (the_left_translation_of (h1,p)) . ((the_left_translation_of (h2,p)) . x) by A9, A13; :: thesis: verum
end;
the_left_translation_of ((h1 * h2),p) in Funcs ((the_sylow_p-subgroups_of_prime (p,G)),(the_sylow_p-subgroups_of_prime (p,G))) by FUNCT_2:9;
then A18: ex f being Function st
( the_left_translation_of ((h1 * h2),p) = f & dom f = the_sylow_p-subgroups_of_prime (p,G) & rng f c= the_sylow_p-subgroups_of_prime (p,G) ) by FUNCT_2:def 2;
now :: thesis: for x being object holds
( ( x in dom (the_left_translation_of ((h1 * h2),p)) implies ( x in dom (the_left_translation_of (h2,p)) & (the_left_translation_of (h2,p)) . x in dom (the_left_translation_of (h1,p)) ) ) & ( x in dom (the_left_translation_of (h2,p)) & (the_left_translation_of (h2,p)) . x in dom (the_left_translation_of (h1,p)) implies x in dom (the_left_translation_of ((h1 * h2),p)) ) )
let x be object ; :: thesis: ( ( x in dom (the_left_translation_of ((h1 * h2),p)) implies ( x in dom (the_left_translation_of (h2,p)) & (the_left_translation_of (h2,p)) . x in dom (the_left_translation_of (h1,p)) ) ) & ( x in dom (the_left_translation_of (h2,p)) & (the_left_translation_of (h2,p)) . x in dom (the_left_translation_of (h1,p)) implies x in dom (the_left_translation_of ((h1 * h2),p)) ) )
assume that
A20: x in dom (the_left_translation_of (h2,p)) and
(the_left_translation_of (h2,p)) . x in dom (the_left_translation_of (h1,p)) ; :: thesis: x in dom (the_left_translation_of ((h1 * h2),p))
thus x in dom (the_left_translation_of ((h1 * h2),p)) by A18, A20; :: thesis: verum
end;
hence H1(h1 * h2) = H1(h1) * H1(h2) by A4, FUNCT_1:10; :: thesis: verum
end;
A21: H1( 1_ H) = id (the_sylow_p-subgroups_of_prime (p,G))
proof
set f = the_left_translation_of ((1_ H),p);
A22: now :: thesis: for x being object st x in the_sylow_p-subgroups_of_prime (p,G) holds
(the_left_translation_of ((1_ H),p)) . x = x
let x be object ; :: thesis: ( x in the_sylow_p-subgroups_of_prime (p,G) implies (the_left_translation_of ((1_ H),p)) . x = x )
assume x in the_sylow_p-subgroups_of_prime (p,G) ; :: thesis: (the_left_translation_of ((1_ H),p)) . x = x
then reconsider P1 = x as Element of the_sylow_p-subgroups_of_prime (p,G) ;
consider P2 being Element of the_sylow_p-subgroups_of_prime (p,G), H1, H2 being strict Subgroup of G, g being Element of G such that
A23: ( P2 = (the_left_translation_of ((1_ H),p)) . P1 & P1 = H1 & P2 = H2 ) and
A24: (1_ H) " = g and
A25: H2 = H1 |^ g by Def20;
(1_ H) " = 1_ H by GROUP_1:8;
then g = 1_ G by A24, GROUP_2:44;
hence (the_left_translation_of ((1_ H),p)) . x = x by A23, A25, GROUP_3:61; :: thesis: verum
end;
thus H1( 1_ H) = id (the_sylow_p-subgroups_of_prime (p,G)) by A22; :: thesis: verum
end;
ex T being LeftOperation of H,(the_sylow_p-subgroups_of_prime (p,G)) st
for h being Element of H holds T . h = H1(h) from GROUP_10:sch 1(A21, A1);
 hence ex b1 being LeftOperation of H,(the_sylow_p-subgroups_of_prime (p,G)) st
for h being Element of H holds b1 . h = the_left_translation_of (h,p) ; :: thesis: verum