let x be object ; :: thesis: for G being non empty multMagma

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 multMagma ; :: 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 )

g in {g} by TARSKI:def 1;

hence x in A * g by A3; :: thesis: verum

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 multMagma ; :: 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

given h being Element of G such that A3:
( x = h * g & h in A )
; :: thesis: x in A * g
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;( 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

g in {g} by TARSKI:def 1;

hence x in A * g by A3; :: thesis: verum