reserve n,n1,n2,m for Nat,
  r,r1,r2,p,g1,g2,g for Real,
  seq,seq9,seq1 for Real_Sequence,
  y for set;

theorem Th1:
  -g<r & r<g iff |.r.|<g
proof
  thus -g<r & r<g implies |.r.|<g
  proof
    assume that
A1: -g<r and
A2: r<g;
A3: |.r.|<=g by A1,A2,ABSVALUE:5;
    now
      assume
A4:   |.r.|=g;
      now
        assume r<0;
        then g=-r by A4,ABSVALUE:def 1;
        hence contradiction by A1;
      end;
      hence contradiction by A2,A4,ABSVALUE:def 1;
    end;
    hence thesis by A3,XXREAL_0:1;
  end;
  assume
A5: |.r.|<g;
  then
A6: -g<=r by ABSVALUE:5;
A7: 0<=|.r.| by COMPLEX1:46;
A8: 0<g by A5,COMPLEX1:46;
A9: -g<-0 by A5,A7;
  now
    assume r=-g;
    then |.r.|=--g by A9,ABSVALUE:def 1
      .=g;
    hence contradiction by A5;
  end;
  hence -g<r by A6,XXREAL_0:1;
  thus thesis by A5,A8,ABSVALUE:def 1;
end;
