let N be Element of NAT ; :: thesis: for seq being Real_Sequence of N holds
( seq is non-zero iff for n being Element of NAT holds seq . n <> 0.REAL N )
let seq be Real_Sequence of N; :: thesis: ( seq is non-zero iff for n being Element of NAT holds seq . n <> 0.REAL N )
thus ( seq is non-zero implies for n being Element of NAT holds seq . n <> 0.REAL N ) by Th3; :: thesis: ( ( for n being Element of NAT holds seq . n <> 0.REAL N ) implies seq is non-zero )
assume for n being Element of NAT holds seq . n <> 0.REAL N ; :: thesis: seq is non-zero
then for x being set st x in NAT holds
seq . x <> 0.REAL N ;
hence seq is non-zero by Th3; :: thesis: verum