let seq, seq1 be Real_Sequence; :: thesis: ( ( seq is non-zero & seq1 is non-zero ) iff seq (#) seq1 is non-zero )

thus ( seq is non-zero & seq1 is non-zero implies seq (#) seq1 is non-zero ) :: thesis: ( seq (#) seq1 is non-zero implies ( seq is non-zero & seq1 is non-zero ) )

thus ( seq is non-zero & seq1 is non-zero implies seq (#) seq1 is non-zero ) :: thesis: ( seq (#) seq1 is non-zero implies ( seq is non-zero & seq1 is non-zero ) )

proof

assume A3:
seq (#) seq1 is non-zero
; :: thesis: ( seq is non-zero & seq1 is non-zero )
assume A1:
( seq is non-zero & seq1 is non-zero )
; :: thesis: seq (#) seq1 is non-zero

end;now :: thesis: for n being Nat holds (seq (#) seq1) . n <> 0

hence
seq (#) seq1 is non-zero
by Th5; :: thesis: verumlet n be Nat; :: thesis: (seq (#) seq1) . n <> 0

A2: (seq (#) seq1) . n = (seq . n) * (seq1 . n) by Th8;

( seq . n <> 0 & seq1 . n <> 0 ) by A1, Th5;

hence (seq (#) seq1) . n <> 0 by A2, XCMPLX_1:6; :: thesis: verum

end;A2: (seq (#) seq1) . n = (seq . n) * (seq1 . n) by Th8;

( seq . n <> 0 & seq1 . n <> 0 ) by A1, Th5;

hence (seq (#) seq1) . n <> 0 by A2, XCMPLX_1:6; :: thesis: verum

now :: thesis: for n being Nat holds seq . n <> 0

hence
seq is non-zero
by Th5; :: thesis: seq1 is non-zero let n be Nat; :: thesis: seq . n <> 0

(seq (#) seq1) . n = (seq . n) * (seq1 . n) by Th8;

hence seq . n <> 0 by A3, Th5; :: thesis: verum

end;(seq (#) seq1) . n = (seq . n) * (seq1 . n) by Th8;

hence seq . n <> 0 by A3, Th5; :: thesis: verum

now :: thesis: for n being Nat holds seq1 . n <> 0

hence
seq1 is non-zero
by Th5; :: thesis: verumlet n be Nat; :: thesis: seq1 . n <> 0

(seq (#) seq1) . n = (seq . n) * (seq1 . n) by Th8;

hence seq1 . n <> 0 by A3, Th5; :: thesis: verum

end;(seq (#) seq1) . n = (seq . n) * (seq1 . n) by Th8;

hence seq1 . n <> 0 by A3, Th5; :: thesis: verum