deffunc H₁( Element of H) -> Function of (Left_Cosets P),(Left_Cosets P) = the_left_translation_of $1,P;
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;
A2: ( the_left_translation_of (h1 * h2),P in Funcs (Left_Cosets P),(Left_Cosets P) & the_left_translation_of h1,P in Funcs (Left_Cosets P),(Left_Cosets P) & the_left_translation_of h2,P in Funcs (Left_Cosets P),(Left_Cosets P) ) by FUNCT_2:12;
then A3: ex f being Function st
( the_left_translation_of h1,P = f & dom f = Left_Cosets P & rng f c= Left_Cosets P ) by FUNCT_2:def 2;
A4: ex f being Function st
( the_left_translation_of h2,P = f & dom f = Left_Cosets P & rng f c= Left_Cosets P ) by A2, FUNCT_2:def 2;
A5: ex f being Function st
( the_left_translation_of (h1 * h2),P = f & dom f = Left_Cosets P & rng f c= Left_Cosets P ) by A2, FUNCT_2:def 2;

A6: now

let x be set ; :: 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) ) )

hereby :: thesis: ( 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 A7: x in dom (the_left_translation_of (h1 * h2),P) ; :: thesis: ( x in dom (the_left_translation_of h2,P) & (the_left_translation_of h2,P) . x in dom (the_left_translation_of h1,P) )
hence x in dom (the_left_translation_of h2,P) by A4; :: thesis: (the_left_translation_of h2,P) . x in dom (the_left_translation_of h1,P)
(the_left_translation_of h2,P) . x in rng (the_left_translation_of h2,P) by A4, A7, FUNCT_1:12;
hence (the_left_translation_of h2,P) . x in dom (the_left_translation_of h1,P) by A3; :: thesis: verum

end;

assume ( x in dom (the_left_translation_of h2,P) & (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)
hence x in dom (the_left_translation_of (h1 * h2),P) by A5; :: thesis: verum

end;

now

let x be set ; :: 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 A8: 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 P1' = x as Element of Left_Cosets P ;
consider P2' being Element of Left_Cosets P, A1', A2' being Subset of G, g' being Element of G such that
A9: ( P2' = (the_left_translation_of h2,P) . P1' & A2' = g' * A1' & A1' = P1' & A2' = P2' & g' = h2 ) by Def8;
(the_left_translation_of h2,P) . x in rng (the_left_translation_of h2,P) by A4, A8, FUNCT_1:12;
then reconsider P1'' = (the_left_translation_of h2,P) . x as Element of Left_Cosets P ;
consider P2'' being Element of Left_Cosets P, A1'', A2'' being Subset of G, g'' being Element of G such that
A10: ( P2'' = (the_left_translation_of h1,P) . P1'' & A2'' = g'' * A1'' & A1'' = P1'' & A2'' = P2'' & g'' = h1 ) by Def8;
reconsider P1''' = x as Element of Left_Cosets P by A8;
consider P2''' being Element of Left_Cosets P, A1''', A2''' being Subset of G, g''' being Element of G such that
A11: ( P2''' = (the_left_translation_of (h1 * h2),P) . P1''' & A2''' = g''' * A1''' & A1''' = P1''' & A2''' = P2''' & g''' = h1 * h2 ) by Def8;
g''' = g'' * g' by A9, A10, A11, GROUP_2:52;
hence (the_left_translation_of (h1 * h2),P) . x = (the_left_translation_of h1,P) . ((the_left_translation_of h2,P) . x) by A9, A10, A11, GROUP_2:38; :: thesis: verum

end;

hence H₁(h1 * h2) = H₁(h1) * H₁(h2) by A6, FUNCT_1:20; :: thesis: verum

end;

A12: H₁( 1_ H) = id (Left_Cosets P)

proof

set f = the_left_translation_of (1_ H),P;
the_left_translation_of (1_ H),P in Funcs (Left_Cosets P),(Left_Cosets P) by FUNCT_2:12;
then A13: ex f' being Function st
( the_left_translation_of (1_ H),P = f' & dom f' = Left_Cosets P & rng f' c= Left_Cosets P ) by FUNCT_2:def 2;

now

let x be set ; :: thesis: ( x in Left_Cosets P implies (the_left_translation_of (1_ H),P) . x = x )
assume x in Left_Cosets P ; :: thesis: (the_left_translation_of (1_ H),P) . x = x
then reconsider P1 = x as Element of Left_Cosets P ;
consider P2 being Element of Left_Cosets P, A1, A2 being Subset of G, g being Element of G such that
A14: ( P2 = (the_left_translation_of (1_ H),P) . P1 & A2 = g * A1 & A1 = P1 & A2 = P2 & g = 1_ H ) by Def8;
1_ H = 1_ G by GROUP_2:53;
hence (the_left_translation_of (1_ H),P) . x = x by A14, GROUP_2:43; :: thesis: verum

end;

hence H₁( 1_ H) = id (Left_Cosets P) by A13, FUNCT_1:34; :: thesis: verum

end;

ex T being LeftOperation of H,(Left_Cosets P) st
for h being Element of H holds T . h = H₁(h) from GROUP_10:sch 1(A12, A1);
hence ex b₁ being LeftOperation of H,(Left_Cosets P) st
for h being Element of H holds b₁ . h = the_left_translation_of h,P ; :: thesis: verum