let P be FinSequence-membered set ; :: thesis: for A being Function of P,NAT
for n, m being Nat holds Polish-expression-hierarchy (P,A,n) c= Polish-expression-hierarchy (P,A,(n + m))
let A be Function of P,NAT; :: thesis: for n, m being Nat holds Polish-expression-hierarchy (P,A,n) c= Polish-expression-hierarchy (P,A,(n + m))
let n, m be Nat; :: thesis: Polish-expression-hierarchy (P,A,n) c= Polish-expression-hierarchy (P,A,(n + m))
defpred S₁[ Nat] means Polish-expression-hierarchy (P,A,n) c= Polish-expression-hierarchy (P,A,(n + $1));
A1: S₁[ 0 ] ;
A2: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A3: S₁[k] ; :: thesis: S₁[k + 1]
Polish-expression-hierarchy (P,A,(n + k)) c= Polish-expression-hierarchy (P,A,((n + k) + 1)) by Th24;
hence S₁[k + 1] by A3, XBOOLE_1:1; :: thesis: verum

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(A1, A2);
hence Polish-expression-hierarchy (P,A,n) c= Polish-expression-hierarchy (P,A,(n + m)) ; :: thesis: verum