let n be Nat; :: thesis: for r being Real st r is irrational holds
(c_d r) . n >= 1
let r be Real; :: thesis: ( r is irrational implies (c_d r) . n >= 1 )
assume A1: r is irrational ; :: thesis: (c_d r) . n >= 1
defpred S1[ Nat] means (c_d r) . $1 >= 1;
A2: S1[ 0 ] by REAL_3:def 6;
A3: for n being Nat st S1[n] holds
S1[n + 1]
proof
let n be Nat; :: thesis: ( S1[n] implies S1[n + 1] )
assume A4: S1[n] ; :: thesis: S1[n + 1]
(c_d r) . (n + 1) >= (c_d r) . n by A1, Th7;
hence S1[n + 1] by A4, XXREAL_0:2; :: thesis: verum
end;
for n being Nat holds S1[n] from NAT_1:sch 2(A2, A3);
 hence (c_d r) . n >= 1 ; :: thesis: verum