let a be Real; :: thesis: ( a <> 0 implies for m being Nat holds (a GeoSeq) . m <> 0 )
assume A1: a <> 0 ; :: thesis: for m being Nat holds (a GeoSeq) . m <> 0
defpred S₁[ Nat] means (a GeoSeq) . $1 <> 0 ;
A2: for n being Nat st S₁[n] holds
S₁[n + 1]

proof

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume A3: (a GeoSeq) . n <> 0 ; :: thesis: S₁[n + 1]
(a GeoSeq) . (n + 1) = ((a GeoSeq) . n) * a by Th3;
hence S₁[n + 1] by A1, A3; :: thesis: verum

end;

A4: S₁[ 0 ] by Th3;
thus for n being Nat holds S₁[n] from NAT_1:sch 2(A4, A2); :: thesis: verum