reserve a,b,i,k,m,n for Nat;
reserve s,z for non zero Nat;
reserve r for Real;
reserve c for Complex;
reserve e1,e2,e3,e4,e5 for ExtReal;

theorem Th32:
  for a,b,c being odd Nat st c-b = 2 & b-a = 2 holds
  3 divides a or 3 divides b or 3 divides c
  proof
    let a,b,c be odd Nat such that
A1: c-b = 2 and
A2: b-a = 2;
    assume
A3: not 3 divides a & not 3 divides b;
A4: (b-a) mod 3 = ((b mod 3)-(a mod 3)) mod 3 by INT_6:7;
    per cases by A3,Lm17;
    suppose a mod 3 = 1 & b mod 3 = 1;
      hence 3 divides c by A2,A4,NAT_D:24,INT_1:73;
    end;
    suppose a mod 3 = 1 & b mod 3 = 2;
      hence 3 divides c by A2,A4,NAT_D:24;
    end;
    suppose that
      a mod 3 = 2 and
A5:   b mod 3 = 1;
A6:   (c-b) mod 3 = ((c mod 3)-(b mod 3)) mod 3 by INT_6:7;
      assume not 3 divides c;
      then per cases by Lm17;
      suppose c mod 3 = 1;
        hence contradiction by A1,A6,A5,NAT_D:24,INT_1:73;
      end;
      suppose c mod 3 = 2;
        hence contradiction by A1,A6,A5,NAT_D:24;
      end;
    end;
    suppose a mod 3 = 2 & b mod 3 = 2;
      hence 3 divides c by A2,A4,INT_1:73;
    end;
  end;
