reserve PM for MetrStruct;
reserve x,y for Element of PM;
reserve r,p,q,s,t for Real;
reserve T for TopSpace;
reserve A for Subset of T;
reserve T for non empty TopSpace;
reserve x for Point of T;
reserve Z,X,V,W,Y,Q for Subset of T;
reserve FX for Subset-Family of T;
reserve a for set;
reserve x,y for Point of T;
reserve A,B for Subset of T;
reserve FX,GX for Subset-Family of T;
reserve x,y,z for Element of PM;
reserve V,W,Y for Subset of PM;

theorem Th27:
  for r being Real st PM is triangle & y in Ball(x,r) holds
  ex p st p>0 & Ball(y,p) c= Ball(x,r)
proof
  let r be Real;
  reconsider r9=r as Real;
  assume
A1: PM is triangle;
  assume
A2: y in Ball(x,r);
  then
A3: dist(x,y) < r by METRIC_1:11;
A4: PM is non empty by A2;
  thus thesis
  proof
    set p = r9 - dist(x,y);
    take p;
    thus p > 0 by A3,XREAL_1:50;
    for z holds z in Ball(y,p) implies z in Ball(x,r)
    proof
      let z;
      assume z in Ball(y,p);
      then dist(y,z) < r9 - dist(x,y) by METRIC_1:11;
      then
A5:   dist(x,y) + dist(y,z) < r by XREAL_1:20;
      dist(x,y) + dist(y,z) >= dist(x,z) by A1,METRIC_1:4;
      then dist(x,z) < r by A5,XXREAL_0:2;
      hence thesis by A4,METRIC_1:11;
    end;
    hence thesis;
  end;
end;
