let G be Group; :: thesis: for H1, H2 being Subgroup of G

for x being Element of G 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: for x being Element of G holds

( x in H1 * H2 iff ex a, b being Element of G st

( x = a * b & a in H1 & b in H2 ) )

let x be Element of G; :: 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 )

( a in carr H1 & b in carr H2 ) by A2, STRUCT_0:def 5;

hence x in H1 * H2 by A2; :: thesis: verum

for x being Element of G 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: for x being Element of G holds

( x in H1 * H2 iff ex a, b being Element of G st

( x = a * b & a in H1 & b in H2 ) )

let x be Element of G; :: 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

given a, b being Element of G such that A2:
( x = a * b & a in H1 & b in H2 )
; :: thesis: x in H1 * H2
assume
x in H1 * H2
; :: thesis: ex a, b being Element of G st

( x = a * b & a in H1 & b in H2 )

then consider a, b being Element of G such that

A1: ( x = a * b & a in carr H1 & b in carr H2 ) ;

( a in H1 & b in H2 ) by A1, STRUCT_0:def 5;

hence ex a, b being Element of G st

( x = a * b & a in H1 & b in H2 ) by A1; :: thesis: verum

end;( x = a * b & a in H1 & b in H2 )

then consider a, b being Element of G such that

A1: ( x = a * b & a in carr H1 & b in carr H2 ) ;

( a in H1 & b in H2 ) by A1, STRUCT_0:def 5;

hence ex a, b being Element of G st

( x = a * b & a in H1 & b in H2 ) by A1; :: thesis: verum

( a in carr H1 & b in carr H2 ) by A2, STRUCT_0:def 5;

hence x in H1 * H2 by A2; :: thesis: verum