reserve n,n1,m,k for Nat;
reserve x,y for set;
reserve s,g,g1,g2,r,p,p2,q,t for Real;
reserve s1,s2,s3 for Real_Sequence;
reserve Nseq for increasing sequence of NAT;
reserve X for Subset of REAL;

theorem Th1:
  r in ].p-g,p+g.[ iff |.r-p.|<g
proof
  thus r in ].p-g,p+g.[ implies |.r-p.|<g
  proof
    assume r in ].p-g,p+g.[;
    then
A1: ex s st r=s & p-g<s & s<p+g;
    then p+-g<r;
    then
A2: -g<r-p by XREAL_1:20;
    r-p<g by A1,XREAL_1:19;
    hence thesis by A2,SEQ_2:1;
  end;
  assume
A3: |.r-p.|<g;
  then r-p<g by SEQ_2:1;
  then
A4: r<p+g by XREAL_1:19;
  -g<r-p by A3,SEQ_2:1;
  then r is Real & p+-g<r by XREAL_1:20;
  hence thesis by A4;
end;
