let a, b, m be Nat; :: thesis: ( a < b implies Ackermann (m,a) < Ackermann (m,b) )
defpred S₁[ Nat] means for a being Nat st a < $1 holds
H₁(m,a) < H₁(m,$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]
let a be Nat; :: thesis: ( a < k + 1 implies H₁(m,a) < H₁(m,k + 1) )
assume a < k + 1 ; :: thesis: H₁(m,a) < H₁(m,k + 1)
then a <= k by NAT_1:13;

per cases then ( a < k or a = k ) by XXREAL_0:1;

suppose a < k ; :: thesis: H₁(m,a) < H₁(m,k + 1)

then H₁(m,a) < H₁(m,k) by A3;
hence H₁(m,a) < H₁(m,k + 1) by Th17, XXREAL_0:2; :: thesis: verum

end;

suppose a = k ; :: thesis: H₁(m,a) < H₁(m,k + 1)

hence H₁(m,a) < H₁(m,k + 1) by Th17; :: thesis: verum

end;

end;

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(A1, A2);
hence ( a < b implies Ackermann (m,a) < Ackermann (m,b) ) ; :: thesis: verum