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 Th51:
  n^2 mod 8 = 0 or n^2 mod 8 = 1 or n^2 mod 8 = 4
  proof
    consider k such that
A1: n = 8*k or n = 8*k+1 or n = 8*k+2 or n = 8*k+3 or
    n = 8*k+4 or n = 8*k+5 or n = 8*k+6 or n = 8*k+7 by NUMBER02:28;
    set a = n mod 8;
A2: n^2 mod 8 = (a*a) mod 8 by NAT_D:67;
    per cases by A1;
    suppose n = 8*k+0;
      hence thesis by A2;
    end;
    suppose n = 8*k+1;
      then a = 1 mod 8 by NAT_D:61
      .= 1 by NAT_D:24;
      hence thesis by A2;
    end;
    suppose n = 8*k+2;
      then a = 2 mod 8 by NAT_D:61
      .= 2 by NAT_D:24;
      hence thesis by A2,NAT_D:24;
    end;
    suppose n = 8*k+3;
      then
A3:   a = 3 mod 8 by NAT_D:61
      .= 3 by NAT_D:24;
      (1+1*8) mod 8 = 1 mod 8 by NAT_D:61;
      hence thesis by A2,A3,NAT_D:24;
    end;
    suppose n = 8*k+4;
      then
A4:   a = 4 mod 8 by NAT_D:61
      .= 4 by NAT_D:24;
      (0+2*8) mod 8 = 0 mod 8;
      hence thesis by A2,A4;
    end;
    suppose n = 8*k+5;
      then
A5:   a = 5 mod 8 by NAT_D:61
      .= 5 by NAT_D:24;
      (1+3*8) mod 8 = 1 mod 8 by NAT_D:61;
      hence thesis by A2,A5,NAT_D:24;
    end;
    suppose n = 8*k+6;
      then
A6:   a = 6 mod 8 by NAT_D:61
      .= 6 by NAT_D:24;
      (4+4*8) mod 8 = 4 mod 8 by NAT_D:61;
      hence thesis by A2,A6,NAT_D:24;
    end;
    suppose n = 8*k+7;
      then
A7:   a = 7 mod 8 by NAT_D:61
      .= 7 by NAT_D:24;
      (1+6*8) mod 8 = 1 mod 8 by NAT_D:61;
      hence thesis by A2,A7,NAT_D:24;
    end;
  end;
