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 Th76:
  for f being Prime, g being Nat st f < g holds SetPrimenumber f
  \/ {f} c= SetPrimenumber g
proof
  let f be Prime,g be Nat;
  assume
A1: f<g;
  let x be object;
  assume
A2: x in SetPrimenumber f\/{f};
  then reconsider x as Nat;
  per cases by A2,XBOOLE_0:def 3;
  suppose
A3: x in SetPrimenumber f;
    then x < f by Def7;
    then
A4: x < g by A1,XXREAL_0:2;
    x is prime by A3,Def7;
    hence thesis by A4,Def7;
  end;
  suppose
    x in {f};
    then x = f by TARSKI:def 1;
    hence thesis by A1,Def7;
  end;
end;
