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

let G be Group; :: thesis: for A being Subset of G
for H being Subgroup of G holds
( x in A * H iff ex g1, g2 being Element of G st
( x = g1 * g2 & g1 in A & g2 in H ) )

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

let H be Subgroup of G; :: thesis: ( x in A * H iff ex g1, g2 being Element of G st
( x = g1 * g2 & g1 in A & g2 in H ) )

thus ( x in A * H implies ex g1, g2 being Element of G st
( x = g1 * g2 & g1 in A & g2 in H ) ) :: thesis: ( ex g1, g2 being Element of G st
( x = g1 * g2 & g1 in A & g2 in H ) implies x in A * H )
proof
assume x in A * H ; :: thesis: ex g1, g2 being Element of G st
( x = g1 * g2 & g1 in A & g2 in H )

then consider g1, g2 being Element of G such that
A1: ( x = g1 * g2 & g1 in A ) and
A2: g2 in carr H ;
g2 in H by A2;
hence ex g1, g2 being Element of G st
( x = g1 * g2 & g1 in A & g2 in H ) by A1; :: thesis: verum
end;
given g1, g2 being Element of G such that A3: ( x = g1 * g2 & g1 in A ) and
A4: g2 in H ; :: thesis: x in A * H
g2 in carr H by A4;
hence x in A * H by A3; :: thesis: verum