let m, n be Nat; :: thesis: for S being Language holds (S -termsOfMaxDepth) . m c= (S -termsOfMaxDepth) . (m + n)
let S be Language; :: thesis: (S -termsOfMaxDepth) . m c= (S -termsOfMaxDepth) . (m + n)
set T = S -termsOfMaxDepth ;
defpred S₁[ Nat] means (S -termsOfMaxDepth) . m c= (S -termsOfMaxDepth) . (m + $1);
A1: S₁[ 0 ] ;
A2: for n being Nat st S₁[n] holds
S₁[n + 1]

proof

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume A3: S₁[n] ; :: thesis: S₁[n + 1]
(S -termsOfMaxDepth) . ((m + n) + 1) = ((S -termsOfMaxDepth) . (m + n)) \/ (union { (Compound (s,((S -termsOfMaxDepth) . (m + n)))) where s is ofAtomicFormula Element of S : s is operational } ) by Def30;
then (S -termsOfMaxDepth) . (m + n) c= (S -termsOfMaxDepth) . ((m + n) + 1) by XBOOLE_1:7;
hence S₁[n + 1] by A3, XBOOLE_1:1; :: thesis: verum

end;

for n being Nat holds S₁[n] from NAT_1:sch 2(A1, A2);
hence (S -termsOfMaxDepth) . m c= (S -termsOfMaxDepth) . (m + n) ; :: thesis: verum