for x being object st x in A holds

x in the carrier of (Lin A)

then reconsider AA = A as finite Subset of (Lin A) ;

Lin AA = Lin A by ZMODUL03:20;

hence Lin A is finitely-generated ; :: thesis: verum

x in the carrier of (Lin A)

proof

then
A c= the carrier of (Lin A)
;
let x be object ; :: thesis: ( x in A implies x in the carrier of (Lin A) )

assume A1: x in A ; :: thesis: x in the carrier of (Lin A)

x in Lin A by A1, MOD_3:5;

hence x in the carrier of (Lin A) ; :: thesis: verum

end;assume A1: x in A ; :: thesis: x in the carrier of (Lin A)

x in Lin A by A1, MOD_3:5;

hence x in the carrier of (Lin A) ; :: thesis: verum

then reconsider AA = A as finite Subset of (Lin A) ;

Lin AA = Lin A by ZMODUL03:20;

hence Lin A is finitely-generated ; :: thesis: verum