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

theorem Th34:
  for r0 being Integer st r0 = numberR0(a,b,c) holds
  a+r0, b+r0, c+r0 at_least_two_are_not_divisible_by 3
  proof
    let r0 be Integer such that
A1: r0 = numberR0(a,b,c);
    per cases;
    suppose
A2:   a,b,c give_three_different_remainders_upon_dividing_by 3;
      then
A3:   a mod 3, b mod 3, c mod 3 are_mutually_distinct;
A4:   r0 = 0 by A1,A2,Def10;
      assume 3 divides a+r0 or 3 divides b+r0 or not 3 divides c+r0;
      hence thesis by A3,A4;
    end;
    suppose
A5:   not a,b,c give_three_different_remainders_upon_dividing_by 3;
      then
A6:   not a mod 3, b mod 3, c mod 3 are_mutually_distinct;
      assume
A7:   not a+r0,b+r0,c+r0 at_least_two_are_not_divisible_by 3;
      then
A8:   (3 divides a+r0 or 3 divides b+r0 or not 3 divides c+r0) &
      (3 divides a+r0 or not 3 divides b+r0 or 3 divides c+r0) &
      (not 3 divides a+r0 or 3 divides b+r0 or 3 divides c+r0) &
      (3 divides a+r0 or 3 divides b+r0 or 3 divides c+r0);
A9:   a+r0 mod 3 = ((a mod 3) + (r0 mod 3)) mod 3 by NAT_D:66;
A10:  b+r0 mod 3 = ((b mod 3) + (r0 mod 3)) mod 3 by NAT_D:66;
A11:  c+r0 mod 3 = ((c mod 3) + (r0 mod 3)) mod 3 by NAT_D:66;
      reconsider a3 = a mod 3 as Element of NAT by INT_1:3,57;
      a mod 3 < 2+1 by INT_1:58;
      then a3 <= 2 by NAT_1:13;
      then
A12:  a3 = 0 or ... or a3 = 2;
      per cases by A6;
      suppose
A13:    a mod 3 = b mod 3;
        then
A14:    r0 = 1 - (a mod 3) by A1,Def10;
        then per cases by A12;
        suppose r0 = 1;
          hence contradiction by A7,A9,A10,A13,A14,Lm5,INT_1:62;
        end;
        suppose r0 = 0;
          hence contradiction by A8,A13,A14;
        end;
        suppose r0 = -1;
          hence contradiction
          by A7,A9,A10,A13,A14,Lm5,Lm6,INT_1:62,NUMBER02:9;
        end;
      end;
      suppose
A15:    a mod 3 = c mod 3;
        then
A16:    r0 = 1 - (a mod 3) by A1,Def10;
        then per cases by A12;
        suppose r0 = 1;
          hence contradiction by A7,A9,A11,A15,A16,Lm5,INT_1:62;
        end;
        suppose r0 = 0;
          hence contradiction by A8,A15,A16;
        end;
        suppose r0 = -1;
          hence contradiction
          by A7,A9,A11,A15,A16,Lm5,Lm6,INT_1:62,NUMBER02:9;
        end;
      end;
      suppose that
A17:    b mod 3 = c mod 3 and
A18:    a mod 3 <> b mod 3 & a mod 3 <> c mod 3;
        reconsider b3 = b mod 3 as Element of NAT by INT_1:3,57;
        b mod 3 < 2+1 by INT_1:58;
        then b3 <= 2 by NAT_1:13;
        then
A19:    b3 = 0 or ... or b3 = 2;
A20:    r0 = 1 - (b mod 3) by A1,A5,A18,Def10;
        then per cases by A19;
        suppose r0 = 1;
          hence contradiction by A7,A10,A11,A17,A20,Lm5,INT_1:62;
        end;
        suppose r0 = 0;
          hence contradiction by A8,A17,A20;
        end;
        suppose r0 = -1;
          hence contradiction
          by A7,A10,A11,A17,A20,Lm5,Lm6,INT_1:62,NUMBER02:9;
        end;
      end;
    end;
  end;
