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
  for c being odd Nat holds x|^2 - y|^3 <> (2*c)|^3 - 1
  proof
A1: x|^2 = x^2 by WSIERP_1:1;
    let c be odd Nat;
    assume
A2: x|^2 - y|^3 = (2*c)|^3 - 1;
A3: (2*c)|^3 = (2*c)*(2*c)*(2*c) by POLYEQ_5:2;
    per cases;
    suppose y is odd;
      then consider s being Integer such that
A4:   y-c = 2*s by ABIAN:11;
A5:   3 mod 8 = 3 by NAT_D:24;
      c^2 mod 8 = 1 by Th60;
      then 3 = ((3 mod 8) * (c^2 mod 8)) mod 8 by NAT_D:24
      .= 3*c^2 mod 8 by NAT_D:67;
      then consider j such that
A6:   3*c^2 = j*8+3 by A5,NAT_D:64,NAT_6:9;
      j*8+3-3 >= 3-3 by A6,XREAL_1:66;
      then j >= 0;
      then reconsider j as Element of NAT by INT_1:3;
      (y-c)^2+3*c^2 = (2*s)^2+(8*j+3) by A4,A6
      .= 4*(s^2+2*j)+3;
      then consider p,q being Nat such that
A7:   p = 4*q+3 and
A8:   p is prime and
A9:   p divides (y-c)^2+3*c^2 by NUMBER05:3;
      y|^3 = y*y*y by POLYEQ_5:2;
      then (y+2*c)*((y-c)^2+3*c^2) = (2*c)|^3 + y|^3 by A3
      .= x|^2 + 1 by A2;
      then (y-c)^2+3*c^2 divides x^2+1 by A1;
      hence thesis by A7,A8,A9,Lm18,INT_2:9;
    end;
    suppose y is even;
      then consider i such that
A10:  y = 2*i by ABIAN:11;
      (2*i)|^3 mod 8 = 0 & (2*c)|^3 mod 8 = 0 by Lm19;
      then (y|^3 + (2*c)|^3) mod 8 = (0+0) mod 8 by A10,NAT_D:66;
      then (y|^3 + (2*c)|^3 - 1) mod 8 = ((0 mod 8) - (1 mod 8)) mod 8
      by INT_6:7
      .= (7+(-1)*8) mod 8 by NAT_D:24
      .= 7 mod 8 by NAT_D:61;
      then x^2 mod 8 = 7 by A1,A2,NAT_D:24;
      hence thesis by Th59,Th60;
    end;
  end;
