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;
reserve p for Prime;

theorem Th45:
  p < 89 implies
  p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or p = 19 or
  p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or p = 47 or p = 53
  or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or p = 79 or p = 83
  proof
    assume p < 89;
    then 1+1 < p+1 & p < 88+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p & p < 83;
      hence thesis by Th43;
    end;
    suppose 83 <= p & p <= 83+1;
      hence thesis by XPRIMES0:84,NAT_1:9;
    end;
    suppose 84 <= p & p <= 84+1;
      hence thesis by XPRIMES0:84,85,NAT_1:9;
    end;
    suppose 85 <= p & p <= 85+1;
      hence thesis by XPRIMES0:85,86,NAT_1:9;
    end;
    suppose 86 <= p & p <= 86+1;
      hence thesis by XPRIMES0:86,87,NAT_1:9;
    end;
    suppose 87 <= p & p <= 87+1;
      hence thesis by XPRIMES0:87,88,NAT_1:9;
    end;
  end;
