reserve a,b,c,d,x,j,k,l,m,n for Nat,
  p,q,t,z,u,v for Integer,
  a1,b1,c1,d1 for Complex;

theorem Th47:
  a>=1 & a|^2 + (a+x)|^2 >= (a+x+1)|^2 implies
  (a+l+1)|^2 + (a+l+1+x)|^2 > (a+l+1+x+1)|^2
  proof
    assume
    A1: a>=1 & a|^2 + (a+x)|^2 >= (a+x+1)|^2;
    defpred P[Nat] means (a+$1+1)|^2 + ((a+$1+1)+x)|^2 > ((a+$1+1)+x+1)|^2;
    A2: P[0] by A1,Th46;
    A3: P[k] implies P[k+1]
    proof
      assume
      A4: P[k];
      (a+k)+1 >= 0+1 by XREAL_1:6;
      hence thesis by A4,Th46;
    end;
    for j holds P[j] from NAT_1:sch 2(A2,A3);
    hence thesis;
  end;
