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

theorem Th70:
  primeNumbers(1,10) = { 2,3,5,7,11 }
  proof
    thus primeNumbers(1,10) c= { 2,3,5,7,11 }
    proof
      let x be object;
      assume
A1:   x in primeNumbers(1,10);
      then x in seq(1,10);
      then consider k being Element of NAT such that
A2:   x = k and
A3:   1+1 <= k & k <= 1+10;
      k = 2 or ... or k = 11 by A3;
      hence thesis by A1,A2,ENUMSET1:def 3,XPRIMES0:4,6,8,9,10,NEWTON:def 6;
    end;
    let x be object;
    assume x in { 2,3,5,7,11 };
    then
A4: x = 2 or x = 3 or x = 5 or x = 7 or x = 11 by ENUMSET1:def 3;
    then
A5: x in SetPrimes by NEWTON:def 6,XPRIMES1:2,3,5,7,11;
    x in seq(1,10) by A4;
    hence thesis by A5,XBOOLE_0:def 4;
  end;
