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 S_{1}[ Nat] means (a GeoSeq) . $1 <> 0 ;

A2: for n being Nat st S_{1}[n] holds

S_{1}[n + 1]
_{1}[ 0 ]
by Th3;

thus for n being Nat holds S_{1}[n]
from NAT_1:sch 2(A4, A2); :: thesis: verum

assume A1: a <> 0 ; :: thesis: for m being Nat holds (a GeoSeq) . m <> 0

defpred S

A2: for n being Nat st S

S

proof

A4:
S
let n be Nat; :: thesis: ( S_{1}[n] implies S_{1}[n + 1] )

assume A3: (a GeoSeq) . n <> 0 ; :: thesis: S_{1}[n + 1]

(a GeoSeq) . (n + 1) = ((a GeoSeq) . n) * a by Th3;

hence S_{1}[n + 1]
by A1, A3; :: thesis: verum

end;assume A3: (a GeoSeq) . n <> 0 ; :: thesis: S

(a GeoSeq) . (n + 1) = ((a GeoSeq) . n) * a by Th3;

hence S

thus for n being Nat holds S