reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve r,s for Real;
reserve p,p1,p2,p3 for Prime;
reserve z for Complex;

theorem
  for x,y,z being Nat holds x^2 - 2*y^2 + 8*z <> 3
  proof
    let x,y,z be Nat;
    assume
A1: x^2 - 2*y^2 + 8*z = 3;
    per cases;
    suppose y is even;
      then consider k such that
A2:   y = 2*k;
A3:   x^2 - 2*(2*k)*(2*k) + 8*z = 3 by A1,A2;
      (8*(k^2-z) + 3) mod 8 = 3 mod 8 by NAT_D:61
      .= 3 by NAT_D:24;
      hence thesis by A3,Th51;
    end;
    suppose y is odd;
      then consider k such that
A4:   y = 2*k+1 by ABIAN:9;
A5:   x^2 - 2*(2*k+1)*(2*k+1) + 8*z = 3 by A1,A4;
      (8*(k^2+k-z) + 5) mod 8 = 5 mod 8 by NAT_D:61
      .= 5 by NAT_D:24;
      hence thesis by A5,Th51;
    end;
  end;
