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 Th7:
  for ra,rb,a,b st ra<rb holds 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 holds ex rc
  st f.rc =d & ra<rc & rc <rb
proof
  let ra,rb,a,b;
  assume
A1: ra<rb;
  then
A2: the carrier of Closed-Interval-TSpace(ra,rb)=[.ra,rb.] by TOPMETR:18;
  let f be continuous Function of Closed-Interval-TSpace(ra,rb),R^1,d;
  assume that
A3: f.ra=a and
A4: f.rb=b and
A5: a>d and
A6: d>b;
A7: dom f=the carrier of Closed-Interval-TSpace(ra,rb) by FUNCT_2:def 1;
A8: [#](Closed-Interval-TSpace(ra,rb)) is connected by A1,Th2;
  now
    assume
A9: for rc st ra<rc & rc <rb holds (f.rc)<>d;
A10: now
      assume d in f.:([#](Closed-Interval-TSpace(ra,rb)));
      then consider x being object such that
A11:  x in dom f and
      x in [#](Closed-Interval-TSpace(ra,rb)) and
A12:  d=f.x by FUNCT_1:def 6;
      x in [.ra,rb.] by A2,A11;
      then reconsider r=x as Real;
      r<=rb by A2,A11,XXREAL_1:1;
      then
A13:  rb>r by A4,A6,A12,XXREAL_0:1;
      ra<=r by A2,A11,XXREAL_1:1;
      then ra<r by A3,A5,A12,XXREAL_0:1;
      hence contradiction by A9,A12,A13;
    end;
    rb in [.ra,rb.] by A1,XXREAL_1:1;
    then
A14: b in f.:([#](Closed-Interval-TSpace(ra,rb))) by A4,A2,A7,FUNCT_1:def 6;
    ra in [.ra,rb.] by A1,XXREAL_1:1;
    then a in f.:([#](Closed-Interval-TSpace(ra,rb))) by A3,A2,A7,FUNCT_1:def 6
;
    hence contradiction by A5,A6,A8,A10,A14,Th3,TOPS_2:61;
  end;
  hence thesis;
end;
