hereby :: thesis: ( ( for x being Nat holds GSq . x is non-multi ) implies GSq is non-multi )

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

end;

assume A2: for x being Nat holds GSq . x is non-multi ; :: thesis: GSq is non-multi
let x be Element of dom GSq; :: according to GLIB_000:def 70 :: thesis: GSq . x is non-multi
thus GSq . x is non-multi by A2; :: thesis: verum