let r, x be Real; :: thesis: for m being Element of NAT st 0 < r holds
(Partial_Sums (Maclaurin cos ,].(- r),r.[,x)) . (2 * m) = (Partial_Sums (x P_cos )) . m
let m be Element of NAT ; :: thesis: ( 0 < r implies (Partial_Sums (Maclaurin cos ,].(- r),r.[,x)) . (2 * m) = (Partial_Sums (x P_cos )) . m )
assume A1: r > 0 ; :: thesis: (Partial_Sums (Maclaurin cos ,].(- r),r.[,x)) . (2 * m) = (Partial_Sums (x P_cos )) . m
defpred S₁[ Element of NAT ] means (Partial_Sums (Maclaurin cos ,].(- r),r.[,x)) . (2 * $1) = (Partial_Sums (x P_cos )) . $1;
A2: S₁[ 0 ]

proof

thus (Partial_Sums (Maclaurin cos ,].(- r),r.[,x)) . (2 * 0 ) = (Maclaurin cos ,].(- r),r.[,x) . (2 * 0 ) by SERIES_1:def 1
.= (((- 1) |^ 0 ) * (x |^ (2 * 0 ))) / ((2 * 0 ) ! ) by A1, Th20
.= (x P_cos ) . 0 by SIN_COS:def 25
.= (Partial_Sums (x P_cos )) . 0 by SERIES_1:def 1 ; :: thesis: verum

end;

A3: for k being Element of NAT st S₁[k] holds
S₁[k + 1]

proof

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume S₁[k] ; :: thesis: S₁[k + 1]
thus (Partial_Sums (Maclaurin cos ,].(- r),r.[,x)) . (2 * (k + 1)) = ((Partial_Sums (Maclaurin cos ,].(- r),r.[,x)) . ((2 * k) + 1)) + ((Maclaurin cos ,].(- r),r.[,x) . (((2 * k) + 1) + 1)) by SERIES_1:def 1
.= ((Partial_Sums (Maclaurin cos ,].(- r),r.[,x)) . ((2 * k) + 1)) + ((((- 1) |^ (k + 1)) * (x |^ (2 * (k + 1)))) / ((2 * (k + 1)) ! )) by A1, Th20
.= ((Partial_Sums (x P_cos )) . k) + ((((- 1) |^ (k + 1)) * (x |^ (2 * (k + 1)))) / ((2 * (k + 1)) ! )) by A1, Th25
.= ((Partial_Sums (x P_cos )) . k) + ((x P_cos ) . (k + 1)) by SIN_COS:def 25
.= (Partial_Sums (x P_cos )) . (k + 1) by SERIES_1:def 1 ; :: thesis: verum

end;

for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A2, A3);
hence (Partial_Sums (Maclaurin cos ,].(- r),r.[,x)) . (2 * m) = (Partial_Sums (x P_cos )) . m ; :: thesis: verum