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

proof

let n be Element of 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 Th4;
hence S₁[n + 1] by A1, A3, XCMPLX_1:6; :: thesis: verum

end;

A4: S₁[ 0 ] by Th4;
thus for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A4, A2); :: thesis: verum