let k be Nat; :: thesis: (@' 1) |^ k = k
defpred S1[ Nat] means (@' 1) |^ $1 = $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 (@' 1) |^ k = k ; :: thesis: S1[k + 1]
then ((@' 1) |^ k) * (@' 1) = k + 1 ;
hence S1[k + 1] by GROUP_1:34; :: thesis: verum
end;
A2: S1[ 0 ] by Th13, GROUP_1:25;
for k being Nat holds S1[k] from NAT_1:sch 2(A2, A1);
 hence (@' 1) |^ k = k ; :: thesis: verum