reserve i,j,k,n,m,l,s,t for Nat;
reserve a,b for Real;
reserve F for real-valued FinSequence;
reserve z for Complex;
reserve x,y for Complex;
reserve r,s,t for natural Number;
reserve p,q for natural Number;
reserve i0,i,i1,i2,i4 for Integer;
reserve x for set;
reserve p for Prime;

theorem Th72:
  for l being Nat ex p being Prime st p > l
proof
  let l be Nat;
  reconsider two = 2 as Prime by INT_2:28;
  reconsider l as Element of NAT by ORDINAL1:def 12;
  l=0 & (ex p st p is prime & p>l) or l=1 & (ex p st p is prime & p>l) or
  2<=l & ex p st p is prime & p>l
  proof
    l <= 2 implies l = 0 or ... or l = 2;
    then per cases;
    case
A1:   l=0;
      take two;
      thus thesis by A1;
    end;
    case
A2:   l=1;
      take two;
      thus thesis by A2;
    end;
    case
A3:   2<=l;
      l<=l! by Th38;
      then 2<=l! by A3,XXREAL_0:2;
      then (2+0) <= (l!+1) by XREAL_1:7;
      then consider j being Element of NAT such that
A4:   j is prime and
A5:   j divides (l!+1) by INT_2:31;
      reconsider j9=j as Prime by A4;
      take j9;
A6:   j<>0 by A4,INT_2:def 4;
      j<>1 by A4,INT_2:def 4;
      hence thesis by A5,A6,Th39,Th41;
    end;
  end;
  hence thesis;
end;
