let x be object ; :: thesis: for G being addGroup
for A being Subset of G
for H being Subgroup of G holds
( x in H + A iff ex g1, g2 being Element of G st
( x = g1 + g2 & g1 in H & g2 in A ) )
let G be addGroup; :: thesis: for A being Subset of G
for H being Subgroup of G holds
( x in H + A iff ex g1, g2 being Element of G st
( x = g1 + g2 & g1 in H & g2 in A ) )
let A be Subset of G; :: thesis: for H being Subgroup of G holds
( x in H + A iff ex g1, g2 being Element of G st
( x = g1 + g2 & g1 in H & g2 in A ) )
let H be Subgroup of G; :: thesis: ( x in H + A iff ex g1, g2 being Element of G st
( x = g1 + g2 & g1 in H & g2 in A ) )
thus ( x in H + A implies ex g1, g2 being Element of G st
( x = g1 + g2 & g1 in H & g2 in A ) ) :: thesis: ( ex g1, g2 being Element of G st
( x = g1 + g2 & g1 in H & g2 in A ) implies x in H + A )

proof

assume x in H + A ; :: thesis: ex g1, g2 being Element of G st
( x = g1 + g2 & g1 in H & g2 in A )
then consider g1, g2 being Element of G such that
A1: x = g1 + g2 and
A2: g1 in carr H and
A3: g2 in A ;
g1 in H by A2;
hence ex g1, g2 being Element of G st
( x = g1 + g2 & g1 in H & g2 in A ) by A1, A3; :: thesis: verum

end;

given g1, g2 being Element of G such that A4: x = g1 + g2 and
A5: g1 in H and
A6: g2 in A ; :: thesis: x in H + A
thus x in H + A by A4, A5, A6; :: thesis: verum