let n be Nat; :: thesis: ( n >= 1 implies n block 2 = (1 / 2) * ((2 |^ n) - 2) )
defpred S₁[ Nat] means $1 block 2 = (1 / 2) * ((2 |^ $1) - 2);
A1: for k being Nat st k >= 1 & S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( k >= 1 & S₁[k] implies S₁[k + 1] )
assume that
A2: k >= 1 and
A3: S₁[k] ; :: thesis: S₁[k + 1]
(k + 1) block 2 = (2 * (k block (1 + 1))) + (k block 1) by Th46
.= (2 * ((1 / 2) * ((2 |^ k) - 2))) + 1 by A2, A3, Th32
.= (1 / 2) * ((2 * (2 |^ k)) - 2)
.= (1 / 2) * ((2 |^ (k + 1)) - 2) by NEWTON:6 ;
hence S₁[k + 1] ; :: thesis: verum

end;

A4: S₁[1] by Th29;
for k being Nat st k >= 1 holds
S₁[k] from NAT_1:sch 8(A4, A1);
hence ( n >= 1 implies n block 2 = (1 / 2) * ((2 |^ n) - 2) ) ; :: thesis: verum