let x be set ; :: thesis: for G being Group
for H1, H2 being Subgroup of G st H1 * H2 = H2 * H1 holds
( x in H1 "\/" H2 iff ex a, b being Element of G st
( x = a * b & a in H1 & b in H2 ) )
let G be Group; :: thesis: for H1, H2 being Subgroup of G st H1 * H2 = H2 * H1 holds
( x in H1 "\/" H2 iff ex a, b being Element of G st
( x = a * b & a in H1 & b in H2 ) )
let H1, H2 be Subgroup of G; :: thesis: ( H1 * H2 = H2 * H1 implies ( x in H1 "\/" H2 iff ex a, b being Element of G st
( x = a * b & a in H1 & b in H2 ) ) )
assume A1: H1 * H2 = H2 * H1 ; :: thesis: ( x in H1 "\/" H2 iff ex a, b being Element of G st
( x = a * b & a in H1 & b in H2 ) )
thus ( x in H1 "\/" H2 implies ex a, b being Element of G st
( x = a * b & a in H1 & b in H2 ) ) :: thesis: ( ex a, b being Element of G st
( x = a * b & a in H1 & b in H2 ) implies x in H1 "\/" H2 )

proof

assume x in H1 "\/" H2 ; :: thesis: ex a, b being Element of G st
( x = a * b & a in H1 & b in H2 )
then x in the carrier of (H1 "\/" H2) by STRUCT_0:def 5;
then x in H1 * H2 by A1, GROUP_4:51;
hence ex a, b being Element of G st
( x = a * b & a in H1 & b in H2 ) by Th4; :: thesis: verum

end;

given a, b being Element of G such that A2: ( x = a * b & a in H1 & b in H2 ) ; :: thesis: x in H1 "\/" H2
x in H1 * H2 by A2, Th4;
then x in the carrier of (H1 "\/" H2) by A1, GROUP_4:51;
hence x in H1 "\/" H2 by STRUCT_0:def 5; :: thesis: verum