let x be object ; :: thesis: for G being non empty addMagma
for A being Subset of G
for g being Element of G holds
( x in g + A iff ex h being Element of G st
( x = g + h & h in A ) )
let G be non empty addMagma ; :: thesis: for A being Subset of G
for g being Element of G holds
( x in g + A iff ex h being Element of G st
( x = g + h & h in A ) )
let A be Subset of G; :: thesis: for g being Element of G holds
( x in g + A iff ex h being Element of G st
( x = g + h & h in A ) )
let g be Element of G; :: thesis: ( x in g + A iff ex h being Element of G st
( x = g + h & h in A ) )
thus ( x in g + A implies ex h being Element of G st
( x = g + h & h in A ) ) :: thesis: ( ex h being Element of G st
( x = g + h & h in A ) implies x in g + A )

proof

assume x in g + A ; :: thesis: ex h being Element of G st
( x = g + h & h in A )
then consider g1, g2 being Element of G such that
A1: x = g1 + g2 and
A2: g1 in {g} and
A3: g2 in A ;
g1 = g by A2, TARSKI:def 1;
hence ex h being Element of G st
( x = g + h & h in A ) by A1, A3; :: thesis: verum

end;

given h being Element of G such that A4: ( x = g + h & h in A ) ; :: thesis: x in g + A
g in {g} by TARSKI:def 1;
hence x in g + A by A4; :: thesis: verum