let x be set ; :: thesis: for G being addGroup
for a being Element of G
for H being Subgroup of G holds
( x in H * a iff ex g being Element of G st
( x = g * a & g in H ) )
let G be addGroup; :: thesis: for a being Element of G
for H being Subgroup of G holds
( x in H * a iff ex g being Element of G st
( x = g * a & g in H ) )
let a be Element of G; :: thesis: for H being Subgroup of G holds
( x in H * a iff ex g being Element of G st
( x = g * a & g in H ) )
let H be Subgroup of G; :: thesis: ( x in H * a iff ex g being Element of G st
( x = g * a & g in H ) )
thus ( x in H * a implies ex g being Element of G st
( x = g * a & g in H ) ) :: thesis: ( ex g being Element of G st
( x = g * a & g in H ) implies x in H * a )

proof

assume x in H * a ; :: thesis: ex g being Element of G st
( x = g * a & g in H )
then x in (carr H) * a by Def6A;
then consider b being Element of G such that
A1: ( x = b * a & b in carr H ) by Th41;
take b ; :: thesis: ( x = b * a & b in H )
thus ( x = b * a & b in H ) by A1; :: thesis: verum

end;

given g being Element of G such that A2: x = g * a and
A3: g in H ; :: thesis: x in H * a
x in (carr H) * a by A2, A3, Th41;
hence x in H * a by Def6A; :: thesis: verum