let r be real number ; :: thesis: ( (scf r) . 1 > 0 implies for n being Nat holds (c_d r) . n > 0 )
assume A1: (scf r) . 1 > 0 ; :: thesis: for n being Nat holds (c_d r) . n > 0
set s = scf r;
set s1 = c_d r;
defpred S2[ Nat] means (c_d r) . $1 > 0 ;
A2: S2[ 0 ] by Def7;
A3: S2[1] by A1, Def7;
A4: 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] )
n + 2 > 1 + 0 by XREAL_1:10;
then A5: (scf r) . (n + 2) >= 0 by Th38;
A6: (c_d r) . (n + 2) = (((scf r) . (n + 2)) * ((c_d r) . (n + 1))) + ((c_d r) . n) by Def7;
assume A7: (c_d r) . n > 0 ; :: thesis: ( not S2[n + 1] or S2[n + 2] )
assume (c_d r) . (n + 1) > 0 ; :: thesis: S2[n + 2]
then ((scf r) . (n + 2)) * ((c_d r) . (n + 1)) >= 0 by A5;
hence S2[n + 2] by A6, A7; :: thesis: verum
end;
A8: for n being Nat holds S2[n] from FIB_NUM:sch 1(A2, A3, A4);
 let n be Nat; :: thesis: (c_d r) . n > 0
thus (c_d r) . n > 0 by A8; :: thesis: verum