let s1, s2 be Real_Sequence; :: thesis: ( s1 . 0 = (scf r) . 0 & ( for n being Nat holds s1 . (n + 1) = (1 / (s1 . n)) + ((scf r) . (n + 1)) ) & s2 . 0 = (scf r) . 0 & ( for n being Nat holds s2 . (n + 1) = (1 / (s2 . n)) + ((scf r) . (n + 1)) ) implies s1 = s2 )
assume that
A3: ( s1 . 0 = (scf r) . 0 & ( for n being Nat holds s1 . (n + 1) = (1 / (s1 . n)) + ((scf r) . (n + 1)) ) ) and
A4: ( s2 . 0 = (scf r) . 0 & ( for n being Nat holds s2 . (n + 1) = (1 / (s2 . n)) + ((scf r) . (n + 1)) ) ) ; :: thesis: s1 = s2
defpred S2[ Nat] means s1 . $1 = s2 . $1;
A5: S2[ 0 ] by A3, A4;
A6: for k being Nat st S2[k] holds
S2[k + 1]
proof
let k be Nat; :: thesis: ( S2[k] implies S2[k + 1] )
assume s1 . k = s2 . k ; :: thesis: S2[k + 1]
hence s1 . (k + 1) = (1 / (s2 . k)) + ((scf r) . (k + 1)) by A3
.= s2 . (k + 1) by A4 ;
:: thesis: verum
end;
A7: for n being Nat holds S2[n] from NAT_1:sch 2(A5, A6);
 let x be Element of NAT ; :: according to FUNCT_2:def 9 :: thesis: s1 . x = s2 . x
thus s1 . x = s2 . x by A7; :: thesis: verum