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