let r be Real; :: thesis: ( (scf r) . 0 > 0 implies for n being Nat holds (c_n r) . n > 0 )
assume A1: (scf r) . 0 > 0 ; :: thesis: for n being Nat holds (c_n r) . n > 0
set s1 = c_n r;
set s = scf r;
defpred S2[ Nat] means (c_n r) . $1 > 0 ;
( (c_n r) . 1 = (((scf r) . 1) * ((scf r) . 0)) + 1 & (scf r) . 1 >= 0 ) by Def5, Th38;
then A2: S2[1] by A1;
let n be Nat; :: thesis: (c_n r) . n > 0
A3: for n being Nat st S2[n] & S2[n + 1] holds
S2[n + 2]
proof
let n be Nat; :: thesis: ( S2[n] & S2[n + 1] implies S2[n + 2] )
assume A4: ( (c_n r) . n > 0 & (c_n r) . (n + 1) > 0 ) ; :: thesis: S2[n + 2]
n + 2 > 1 + 0 by XREAL_1:8;
then A5: (scf r) . (n + 2) >= 0 by Th38;
(c_n r) . (n + 2) = (((scf r) . (n + 2)) * ((c_n r) . (n + 1))) + ((c_n r) . n) by Def5;
hence S2[n + 2] by A5, A4; :: thesis: verum
end;
A6: S2[ 0 ] by A1, Def5;
for n being Nat holds S2[n] from FIB_NUM:sch 1(A6, A2, A3);
 hence (c_n r) . n > 0 ; :: thesis: verum