reserve x,y,z for set;
reserve f,f1,f2,f3 for FinSequence,
  p,p1,p2,p3 for set,
  i,k for Nat;
reserve D for non empty set,
  p,p1,p2,p3 for Element of D,
  f,f1,f2 for FinSequence of D;
reserve D for non empty set;
reserve p, q for FinSequence,
  X, Y, x, y for set,
  D for non empty set,
  i, j, k, l, m, n, r for Nat;

theorem Th1:
  for m,k,n being Nat holds m+1<=k & k<=n iff
   ex i being Nat st m<=i & i<n & k=i+1
proof
  let m,k,n be Nat;
  hereby
    reconsider a19=1 as Integer;
    reconsider k9=k as Integer;
    assume that
A1: m+1<=k and
A2: k<=n;
    1<=m+1 by NAT_1:11;
    then k9-a19 in NAT by A1,INT_1:5,XXREAL_0:2;
    then reconsider i=k9-a19 as Nat;
    take i;
    m+1-1<=k-1 by A1,XREAL_1:9;
    hence m<=i;
    k<k+1 by NAT_1:13;
    then k-1<k+1-1 by XREAL_1:9;
    hence i<n by A2,XXREAL_0:2;
    thus k=i+1;
  end;
  given i being Nat such that
A3: m<=i and
A4: i<n and
A5: k=i+1;
  thus thesis by A3,A4,A5,NAT_1:13,XREAL_1:7;
end;
