reserve a,b,c,k,m,n for Nat;
reserve i,j,x,y for Integer;
reserve p,q for Prime;
reserve r,s for Real;

theorem
  x|^2 - y|^3 <> 7
  proof
    assume
A1: x|^2 - y|^3 = 7;
A2: x|^2 = x^2 by WSIERP_1:1;
A3: y|^3 = y*y*y by POLYEQ_5:2;
    per cases;
    suppose y is even;
      then consider i such that
A4:   y = 2*i by ABIAN:11;
      i|^3 = i*i*i by POLYEQ_5:2;
      then
A5:   y|^3 = 8*i|^3 + 0 by A3,A4;
A6:   x^2 mod 8 = 0 or x^2 mod 8 = 1 or x^2 mod 8 = 4 by Th59,Th60;
      (x|^2 - 7) mod 8 = (x|^2-7+8*1) mod 8 by NAT_D:61;
      then (x|^2 - 7) mod 8 = (x|^2 + 1) mod 8
      .= ((x|^2 mod 8) + (1 mod 8)) mod 8 by NAT_D:66;
      then (x|^2 - 7) mod 8 = (0+1) mod 8 or  (x|^2 - 7) mod 8 = (1+1) mod 8
      or (x|^2 - 7) mod 8 = (4+1) mod 8 by A2,A6,NAT_D:24;
      hence thesis by A1,A5,NAT_D:24;
    end;
    suppose y is odd;
      then consider k being Integer such that
A7:   y = 2*k+1 by ABIAN:1;
      x|^2+1 = y|^3+8 by A1
      .= (y+2)*((y-1)^2+3) by A3;
      then
A8:   4*k^2+3 divides x|^2+1 by A7;
      ex p,q being Nat st p = 4*q+3 & p is prime & p divides 4*k^2+3
      by NUMBER05:3;
      hence thesis by A2,A8,INT_2:9,Lm18;
    end;
  end;
