reserve a,b,c,k,m,n for Nat;
reserve i,j,x,y for Integer;
reserve p,q for Prime;
reserve r,s for Real;

theorem
  <=6n+1(1) = {0,1,2,3,4,5,6,7}
  proof
    set A = <=6n+1(1);
    set B = {0,1,2,3,4,5,6,7};
    let x be Nat;
    hereby
      assume x in A;
      then consider a being Nat such that
A1:   a = x and
A2:   a <= 6*1+1;
      a = 0 or ... or a = 7 by A2;
      hence x in B by A1,ENUMSET1:def 6;
    end;
    assume x in B;
    then
    x = 0 or x = 1 or x = 2 or x = 3 or x = 4 or x = 5 or x = 6 or x = 7
    by ENUMSET1:def 6;
    hence thesis;
  end;
