reserve x for set;
reserve a,b,d,ra,rb,r0,s1,s2 for Real;
reserve r,s,r1,r2,r3,rc for Real;
reserve p,q,q1,q2 for Point of TOP-REAL 2;
reserve X,Y,Z for non empty TopSpace;

theorem ::The Bolzano Theorem
  for ra,rb for g being continuous Function of Closed-Interval-TSpace(ra
,rb),R^1,s1,s2 st ra<rb & s1*s2<0 & s1=g.ra & s2=g.rb ex r1 st g.r1=0 & ra<r1 &
  r1<rb
proof
  let ra,rb;
  let g be continuous Function of Closed-Interval-TSpace(ra,rb),R^1,s1,s2;
  assume that
A1: ra<rb and
A2: s1*s2<0;
  s1>0 & s2<0 or s1<0 & s2>0 by A2,XREAL_1:133;
  hence thesis by A1,Th6,Th7;
end;
