let x be object ; :: thesis: for G being Group
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 Group; :: 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
g1 in carr H by A5;
hence x in H * A by A4, A6; :: thesis: verum