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

proof

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

end;

given h being Element of G such that A3: ( x = h + g & h in A ) ; :: thesis: x in A + g
g in {g} by TARSKI:def 1;
hence x in A + g by A3; :: thesis: verum