reserve a,b,c,h for Integer;
reserve k,m,n for Nat;
reserve i,j,z for Integer;
reserve p for Prime;

theorem Th92:
  (a^2+b^2+c^2) mod 8 = 0 or (a^2+b^2+c^2) mod 8 = 1 or (a^2+b^2+c^2) mod 8 = 2
  or
  (a^2+b^2+c^2) mod 8 = 3 or (a^2+b^2+c^2) mod 8 = 4 or (a^2+b^2+c^2) mod 8 = 5
  or
  (a^2+b^2+c^2) mod 8 = 6
  proof
    set x = a^2+b^2+c^2;
A1: x mod 8 = ((a^2 mod 8)+(b^2 mod 8)+(c^2 mod 8)) mod 8
    by NUMBER05:8;
A2: (1*8+1) mod 8 = 1 mod 8 by NAT_D:21
    .= 1 by NAT_D:24;
    now
      per cases by NUMBER10:61;
      case
A3:     a^2 mod 8 = 0;
        per cases by NUMBER10:61;
        case
A4:       b^2 mod 8 = 0;
          per cases by NUMBER10:61;
          case c^2 mod 8 = 0;
            hence x mod 8 = 0 by A1,A3,A4;
          end;
          case c^2 mod 8 = 1;
            hence x mod 8 = 1 by A1,A3,A4;
          end;
          case c^2 mod 8 = 4;
            hence x mod 8 = 4 by A1,A3,A4;
          end;
        end;
        case
A5:       b^2 mod 8 = 1;
          per cases by NUMBER10:61;
          case c^2 mod 8 = 0;
            hence x mod 8 = 1 by A1,A3,A5;
          end;
          case c^2 mod 8 = 1;
            hence x mod 8 = 2 by A1,A3,A5,NAT_D:24;
          end;
          case c^2 mod 8 = 4;
            hence x mod 8 = 5 by A1,A3,A5,NAT_D:24;
          end;
        end;
        case
A6:       b^2 mod 8 = 4;
          per cases by NUMBER10:61;
          case c^2 mod 8 = 0;
            hence x mod 8 = 4 by A1,A3,A6;
          end;
          case c^2 mod 8 = 1;
            hence x mod 8 = 5 by A1,A3,A6,NAT_D:24;
          end;
          case c^2 mod 8 = 4;
            hence x mod 8 = 0 by A1,A3,A6,NAT_D:25;
          end;
        end;
      end;
      case
A7:     a^2 mod 8 = 1;
        per cases by NUMBER10:61;
        case
A8:       b^2 mod 8 = 0;
          per cases by NUMBER10:61;
          case c^2 mod 8 = 0;
            hence x mod 8 = 1 by A1,A7,A8;
          end;
          case c^2 mod 8 = 1;
            hence x mod 8 = 2 by A1,A7,A8,NAT_D:24;
          end;
          case c^2 mod 8 = 4;
            hence x mod 8 = 5 by A1,A7,A8,NAT_D:24;
          end;
        end;
        case
A9:       b^2 mod 8 = 1;
          per cases by NUMBER10:61;
          case c^2 mod 8 = 0;
            hence x mod 8 = 2 by A1,A7,A9,NAT_D:24;
          end;
          case c^2 mod 8 = 1;
            hence x mod 8 = 3 by A1,A7,A9,NAT_D:24;
          end;
          case c^2 mod 8 = 4;
            hence x mod 8 = 6 by A1,A7,A9,NAT_D:24;
          end;
        end;
        case
A10:       b^2 mod 8 = 4;
          per cases by NUMBER10:61;
          case c^2 mod 8 = 0;
            hence x mod 8 = 5 by A1,A7,A10,NAT_D:24;
          end;
          case c^2 mod 8 = 1;
            hence x mod 8 = 6 by A1,A7,A10,NAT_D:24;
          end;
          case c^2 mod 8 = 4;
            hence x mod 8 = 1 by A2,A1,A7,A10;
          end;
        end;
      end;
      case
A11:     a^2 mod 8 = 4;
        per cases by NUMBER10:61;
        case
A12:       b^2 mod 8 = 0;
          per cases by NUMBER10:61;
          case c^2 mod 8 = 0;
            hence x mod 8 = 4 by A1,A11,A12;
          end;
          case c^2 mod 8 = 1;
            hence x mod 8 = 5 by A1,A11,A12,NAT_D:24;
          end;
          case c^2 mod 8 = 4;
            hence x mod 8 = 0 by A1,A11,A12,NAT_D:25;
          end;
        end;
        case
A13:       b^2 mod 8 = 1;
          per cases by NUMBER10:61;
          case c^2 mod 8 = 0;
            hence x mod 8 = 5 by A1,A11,A13,NAT_D:24;
          end;
          case c^2 mod 8 = 1;
            hence x mod 8 = 6 by A1,A11,A13,NAT_D:24;
          end;
          case c^2 mod 8 = 4;
            hence x mod 8 = 1 by A2,A1,A11,A13;
          end;
        end;
        case
A14:       b^2 mod 8 = 4;
          per cases by NUMBER10:61;
          case c^2 mod 8 = 0;
            hence x mod 8 = 0 by A1,A11,A14,NAT_D:25;
          end;
          case c^2 mod 8 = 1;
            hence x mod 8 = 1 by A2,A1,A11,A14;
          end;
          case c^2 mod 8 = 4;
            hence x mod 8 = (1*8+4) mod 8 by A1,A11,A14
            .= 4 mod 8 by NAT_D:21
            .= 4 by NAT_D:24;
          end;
        end;
      end;
    end;
    hence thesis;
  end;
