let x be object ; :: 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 consider g being Element of G such that
A1: ( x = g + a & g in carr H ) by Th28;
take g ; :: thesis: ( x = g + a & g in H )
thus ( x = g + a & g 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
thus x in H + a by A2, A3, Th28; :: thesis: verum