set F = { (g ") where g is Element of G : g in A } ;

{ (g ") where g is Element of G : g in A } c= the carrier of G

{ (g ") where g is Element of G : g in A } c= the carrier of G

proof

hence
{ (g ") where g is Element of G : g in A } is Subset of G
; :: thesis: verum
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in { (g ") where g is Element of G : g in A } or x in the carrier of G )

assume x in { (g ") where g is Element of G : g in A } ; :: thesis: x in the carrier of G

then ex g being Element of G st

( x = g " & g in A ) ;

hence x in the carrier of G ; :: thesis: verum

end;assume x in { (g ") where g is Element of G : g in A } ; :: thesis: x in the carrier of G

then ex g being Element of G st

( x = g " & g in A ) ;

hence x in the carrier of G ; :: thesis: verum