reserve n,m,k for Nat;
reserve r,r1 for Real;
reserve f,seq,seq1 for Real_Sequence;
reserve x,y for set;
reserve e1,e2 for ExtReal;
reserve Nseq for increasing sequence of NAT;
reserve v for FinSequence of REAL,
  r,s for Real,
  n,m,i,j,k for Nat;

theorem
  n>1 iff ex m st n=m+1 & m>0
proof
  thus n>1 implies ex m st n=m+1 & m>0
  proof
    assume
A1: 1<n;
    then consider m being Nat such that
A2: n = m+1 by NAT_1:6;
    reconsider m as Element of NAT by ORDINAL1:def 12;
    take m;
    m <> 0 by A1,A2;
    hence thesis by A2;
  end;
  given m such that
A3: n=m+1 & m>0;
  0+1<n by A3,XREAL_1:6;
  hence thesis;
end;
