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

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

end;

assume A2: for x being Nat holds GSq . x is loopless ; :: thesis: GSq is loopless
let x be Element of dom GSq; :: according to GLIB_000:def 67 :: thesis: GSq . x is loopless
thus GSq . x is loopless by A2; :: thesis: verum