let n be Nat; :: thesis: for R1, R2 being Real_Sequence st ( for i being Nat st i <= n holds
R1 . i = R2 . i ) holds
Sum R1,n = Sum R2,n
let R1, R2 be Real_Sequence; :: thesis: ( ( for i being Nat st i <= n holds
R1 . i = R2 . i ) implies Sum R1,n = Sum R2,n )
A1: CastNat n = n by MODELC_2:def 1;
defpred S₁[ Nat] means ( ( for i being Nat st i <= $1 holds
R1 . i = R2 . i ) implies Sum R1,(CastNat $1) = Sum R2,(CastNat $1) );
A2: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A3: S₁[k] ; :: thesis: S₁[k + 1]
S₁[k + 1]

proof

set m = k + 1;
assume A4: for i being Nat st i <= k + 1 holds
R1 . i = R2 . i ; :: thesis: Sum R1,(CastNat (k + 1)) = Sum R2,(CastNat (k + 1))
then A5: ( CastNat k = k & R1 . (k + 1) = R2 . (k + 1) ) by MODELC_2:def 1;
A6: for i being Nat st i <= k holds
R1 . i = R2 . i

proof

A7: k <= k + 1 by NAT_1:11;
let i be Nat; :: thesis: ( i <= k implies R1 . i = R2 . i )
assume i <= k ; :: thesis: R1 . i = R2 . i
hence R1 . i = R2 . i by A4, A7, XXREAL_0:2; :: thesis: verum

end;

reconsider k = k as Element of NAT by ORDINAL1:def 13;
Sum R1,(CastNat (k + 1)) = Sum R1,(k + 1) by MODELC_2:def 1
.= (Sum R2,k) + (R2 . (k + 1)) by A3, A6, A5, SERIES_1:def 1
.= Sum R2,(k + 1) by SERIES_1:def 1 ;
hence Sum R1,(CastNat (k + 1)) = Sum R2,(CastNat (k + 1)) by MODELC_2:def 1; :: thesis: verum

end;

hence S₁[k + 1] ; :: thesis: verum

end;

A8: S₁[ 0 ]

proof

assume A9: for i being Nat st i <= 0 holds
R1 . i = R2 . i ; :: thesis: Sum R1,(CastNat 0 ) = Sum R2,(CastNat 0 )
Sum R1,(CastNat 0 ) = Sum R1,0 by MODELC_2:def 1
.= R1 . 0 by SERIES_1:def 1
.= R2 . 0 by A9
.= Sum R2,0 by SERIES_1:def 1 ;
hence Sum R1,(CastNat 0 ) = Sum R2,(CastNat 0 ) by MODELC_2:def 1; :: thesis: verum

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(A8, A2);
hence ( ( for i being Nat st i <= n holds
R1 . i = R2 . i ) implies Sum R1,n = Sum R2,n ) by A1; :: thesis: verum