reserve n,k,k1,m,m1,n1,n2,l for Nat;
reserve r,r1,r2,p,p1,g,g1,g2,s,s1,s2,t for Real;
reserve seq,seq1,seq2 for Real_Sequence;
reserve Nseq for increasing sequence of NAT;
reserve x for set;
reserve X,Y for Subset of REAL;

theorem Th2:
  0<p & (ex r st r in X) & (for r st r in X holds r+p in X) implies
  for g ex r st r in X & g<r
proof
  assume that
A1: 0<p and
A2: ex r st r in X and
A3: for r st r in X holds r+p in X and
A4: ex g st for r st r in X holds not g<r;
  defpred X[Real] means for r st r in X holds not $1<r;
  consider Y such that
A5: for p1 being Element of REAL holds p1 in Y iff X[p1] from SUBSET_1:sch 3;
  now
    let r,p1 such that
A6: r in X and
A7: p1 in Y;
    r+p in X by A3,A6;
    then
A8: r+p<=p1 by A5,A7;
    r+(0 qua Nat)<r+p by A1,XREAL_1:8;
    hence r<p1 by A8,XXREAL_0:2;
  end;
  then consider g2 such that
A9: for r,p1 st r in X & p1 in Y holds r<=g2 & g2<=p1 by Th1;
  consider g1 such that
A10: for r st r in X holds not g1<r by A4;
  g1 in REAL by XREAL_0:def 1;
  then
A11: g1 in Y by A10,A5;
  reconsider g2p = g2-p as Element of REAL by XREAL_0:def 1;
A12: now
    assume not g2-p in Y;
    then not X[g2p] by A5;
    then consider r1 being Real such that
A13: r1 in X & g2p<r1;
    g2-p+p<r1+p & r1+p in X by A3,A13,XREAL_1:6;
    hence contradiction by A11,A9;
  end;
  -p<-(0 qua Nat) by A1;
  then g2+-p<g2+(0 qua Nat) by XREAL_1:6;
  hence contradiction by A2,A9,A12;
end;
