deffunc H₁( Element of NAT ) -> Event of = (A1 . $1) ` ;
consider f being sequence of (bool X) such that
A1: for x being Element of NAT holds f . x = H<sub>1</sub>(x) from FUNCT_2:sch 4();
A2: for x being Nat holds f . x = (A1 . x) `

proof

let x be Nat; :: thesis: f . x = (A1 . x) `
reconsider x = x as Element of NAT by ORDINAL1:def 12;
f . x = H<sub>1</sub>(x) by A1;
hence f . x = (A1 . x) ` ; :: thesis: verum

end;

take f ; :: thesis: for n being Nat holds f . n = (A1 . n) `
thus for n being Nat holds f . n = (A1 . n) ` by A2; :: thesis: verum