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
  r in [.p,g.] iff |.p+g-2*r.|<=g-p
proof
  thus r in [.p,g.] implies |.p+g-2*r.|<=g-p
  proof
    assume r in [.p,g.];
    then
A1: ex s st s=r & p<=s & s<=g;
    then 2*r<=2*g by XREAL_1:64;
    then -2*r>=-2*g by XREAL_1:24;
    then (p+g)+-2*r>=(p+g)+-2*g by XREAL_1:7;
    then
A2: p+g-2*r>=-(g-p);
    2*p<=2*r by A1,XREAL_1:64;
    then -2*p>=-2*r by XREAL_1:24;
    then (p+g)+-2*p>=(p+g)+-2*r by XREAL_1:7;
    hence thesis by A2,ABSVALUE:5;
  end;
  assume
A3: |.p+g-2*r.|<=g-p;
  then p+g-2*r>=-(g-p) by ABSVALUE:5;
  then p+g>=p-g+2*r by XREAL_1:19;
  then p+g-(p-g)>=2*r by XREAL_1:19;
  then
A4: 1/2*(2*g)>=1/2*(2*r) by XREAL_1:64;
  g-p>=p+g-2*r by A3,ABSVALUE:5;
  then 2*r+(g-p)>=p+g by XREAL_1:20;
  then 2*r>=p+g-(g-p) by XREAL_1:20;
  then 1/2*(2*r)>=1/2*(2*p) by XREAL_1:64;
 hence r in [.p,g.] by A4;
end;
