reserve x,X for set;
reserve x0,r1,r2,g,g1,g2,p,s for Real;
reserve r for Real;
reserve n,m for Nat;
reserve a,b,d for Real_Sequence;
reserve f for PartFunc of REAL,REAL;

theorem Th3:
  g = x0 + r1 & |.r1.| < r implies 0 < r & g in ].x0 - r,x0 + r.[
proof
  assume that
A1: g = x0 + r1 and
A2: |.r1.| < r;
  thus 0 < r by A2,COMPLEX1:46;
  |.g - x0.| < r by A1,A2;
  hence thesis by RCOMP_1:1;
end;
