
theorem
  for a, b, c, d being Real, f being Function of Closed-Interval-TSpace(
  a,b), Closed-Interval-TSpace(c,d), P being non empty Subset of
Closed-Interval-TSpace(a,b), PP, QQ being Subset of R^1 st a < b & c < d & PP =
P & f is continuous one-to-one & PP is compact & f.a = c & f.b = d & f.:P = QQ
  holds f.(lower_bound [#]PP) = lower_bound [#]QQ
proof
  let a, b, c, d be Real;
  let f be Function of Closed-Interval-TSpace(a,b), Closed-Interval-TSpace(c,d
  ), P be non empty Subset of Closed-Interval-TSpace(a,b), PP, QQ be Subset of
  R^1;
  assume that
A1: a < b & c < d and
A2: PP = P and
A3: f is continuous and
A4: f is one-to-one and
A5: PP is compact and
A6: f.a = c & f.b = d and
A7: f.:P = QQ;
A8: [#] Closed-Interval-TSpace(c,d) = the carrier of Closed-Interval-TSpace(
  c,d);
  set IT = f.(lower_bound [#]PP);
A9: [#]PP is real-bounded by A5,WEIERSTR:11;
  [#]PP <> {} by A2,WEIERSTR:def 1;
  then
A10: lower_bound [#]PP in [#]PP by A5,A9,RCOMP_1:13,WEIERSTR:12;
  then
A11: lower_bound [#]PP in P by A2,WEIERSTR:def 1;
  P c= the carrier of Closed-Interval-TSpace(a,b);
  then
A12: [#]PP c= the carrier of Closed-Interval-TSpace(a,b) by A2,WEIERSTR:def 1;
  reconsider IT as Real;
A13: for r be Real st r in [#]QQ holds IT<=r
  proof
    let r be Real;
    assume r in [#]QQ;
    then r in f.:P by A7,WEIERSTR:def 1;
    then r in f.:[#]PP by A2,WEIERSTR:def 1;
    then consider x be object such that
A14: x in dom f and
A15: x in [#]PP and
A16: r = f.x by FUNCT_1:def 6;
    reconsider x1 = x, x2 = lower_bound [#]PP as Point of
    Closed-Interval-TSpace(a,b) by A11,A14;
    x1 in the carrier of Closed-Interval-TSpace(a,b);
    then reconsider r1 = x, r2 = x2 as Real;
A17: r2 <= r1 by A9,A15,SEQ_4:def 2;
    reconsider fr = f.x2, fx = f.x1 as Real;
    per cases;
    suppose
      r2 <> r1;
      then r2 < r1 by A17,XXREAL_0:1;
      then fr < fx by A1,A3,A4,A6,Th15;
      hence thesis by A16;
    end;
    suppose
      r2 = r1;
      hence thesis by A16;
    end;
  end;
  [#]Closed-Interval-TSpace(a,b) = the carrier of Closed-Interval-TSpace( a,b);
  then P is compact by A2,A5,COMPTS_1:2;
  then
  for P1 being Subset of Closed-Interval-TSpace(c,d) st P1=QQ holds P1 is
  compact by A3,A7,WEIERSTR:8;
  then QQ is compact by A7,A8,COMPTS_1:2;
  then
A18: [#]QQ is real-bounded by WEIERSTR:11;
  lower_bound [#]PP in the carrier of Closed-Interval-TSpace(a,b) by A12,A10;
  then lower_bound [#]PP in dom f by FUNCT_2:def 1;
  then IT in QQ by A7,A11,FUNCT_1:def 6;
  then
A19: IT in [#]QQ by WEIERSTR:def 1;
  for s be Real st 0<s ex r be Real st r in [#]QQ & r<IT+s
  proof
    given s be Real such that
A20: 0<s and
A21: not ex r be Real st r in [#]QQ & r<IT+s;
    IT + 0 < IT + s by A20,XREAL_1:8;
    hence thesis by A19,A21;
  end;
  hence thesis by A18,A19,A13,SEQ_4:def 2;
end;
