hereby :: thesis: ( ( for x being Nat holds GSq . x is simple ) implies GSq is simple )

assume A1: GSq is simple ; :: thesis: for x being Nat holds GSq . x is simple
let x be Nat; :: thesis: GSq . x is simple
x in dom GSq by LmNat;
hence GSq . x is simple by A1; :: thesis: verum

end;

assume A2: for x being Nat holds GSq . x is simple ; :: thesis: GSq is simple
let x be Element of dom GSq; :: according to GLIB_000:def 72 :: thesis: GSq . x is simple
thus GSq . x is simple by A2; :: thesis: verum