defpred S₁[ Nat] means ( 0 in dyadic $1 & 1 in dyadic $1 );
A1: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A2: ( 0 in dyadic k & 1 in dyadic k ) ; :: thesis: S₁[k + 1]
dyadic k c= dyadic (k + 1) by Th5;
hence S₁[k + 1] by A2; :: thesis: verum

end;

A3: S₁[ 0 ] by Th2, TARSKI:def 2;
for k being Nat holds S₁[k] from NAT_1:sch 2(A3, A1);
hence for n being Nat holds
( 0 in dyadic n & 1 in dyadic n ) ; :: thesis: verum