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 Th6:
  for ra,rb,a,b st ra<rb for f being continuous Function of
Closed-Interval-TSpace(ra,rb),R^1,d st f.ra=a & f.rb=b & a<d & d<b ex rc st f.
  rc =d & ra<rc & rc <rb
proof
  let ra,rb,a,b;
  assume
A1: ra<rb;
  let f be continuous Function of Closed-Interval-TSpace(ra,rb),R^1,d;
  assume that
A2: f.ra=a and
A3: f.rb=b and
A4: a<d and
A5: d<b;
  now
    reconsider C=f.:([#](Closed-Interval-TSpace(ra,rb))) as Subset of R^1;
A6: dom f=the carrier of Closed-Interval-TSpace(ra,rb) by FUNCT_2:def 1;
A7: the carrier of Closed-Interval-TSpace(ra,rb)=[.ra,rb.] by A1,TOPMETR:18;
    then rb in [#](Closed-Interval-TSpace(ra,rb)) by A1,XXREAL_1:1;
    then
A8: b in f.:([#](Closed-Interval-TSpace(ra,rb))) by A3,A6,FUNCT_1:def 6;
    assume
A9: not ex rc st (f.rc) =d & ra<rc & rc <rb;
A10: now
      assume d in f.:([#](Closed-Interval-TSpace(ra,rb)));
      then consider x being object such that
A11:  x in the carrier of Closed-Interval-TSpace(ra,rb) and
      x in [#](Closed-Interval-TSpace(ra,rb)) and
A12:  d = f.x by FUNCT_2:64;
      reconsider r=x as Real by A11;
      r<=rb by A7,A11,XXREAL_1:1;
      then
A13:  r<rb by A3,A5,A12,XXREAL_0:1;
      ra<=r by A7,A11,XXREAL_1:1;
      then ra<r by A2,A4,A12,XXREAL_0:1;
      hence contradiction by A9,A12,A13;
    end;
    ra in [#](Closed-Interval-TSpace(ra,rb)) by A1,A7,XXREAL_1:1;
    then a in f.:([#](Closed-Interval-TSpace(ra,rb))) by A2,A6,FUNCT_1:def 6;
    then not C is connected by A4,A5,A10,A8,Th3;
    hence contradiction by A1,Th2,TOPS_2:61;
  end;
  hence thesis;
end;
