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: S₁[ 0 ] by Th4;
A3: 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 A4: (a GeoSeq ) . n <> 0 ; :: thesis: S₁[n + 1]
(a GeoSeq ) . (n + 1) = ((a GeoSeq ) . n) * a by Th4;
hence S₁[n + 1] by A1, A4, XCMPLX_1:6; :: thesis: verum

end;

thus for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A2, A3); :: thesis: verum