reserve a,b,c,d for Real;

theorem Th8:
  a <= b implies for t1,t2 being Point of Closed-Interval-TSpace(a,b) holds
  L[01](t1,t2) is continuous
proof
  assume
A1: a <= b;
  let t1,t2 be Point of Closed-Interval-TSpace(a,b);
  reconsider r1 = t1, r2 = t2 as Real;
  deffunc U(Real) = In((r2 - r1)*$1 + r1,REAL);
  consider L being Function of REAL,REAL such that
A2: for r being Element of REAL holds L.r= U(r) from FUNCT_2:sch 4;
A3: for r being Real holds L.r= (r2 - r1)*r + r1
    proof let r be Real;
      reconsider r as Element of REAL by XREAL_0:def 1;
      L.r= U(r) by A2;
     hence thesis;
    end;
  reconsider f = L as Function of R^1, R^1 by TOPMETR:17;
A4: for s being Point of Closed-Interval-TSpace(0,1), w being Point of R^1
  st s = w holds L[01](t1,t2).s = f.w
  proof
    let s be Point of Closed-Interval-TSpace(0,1), w be Point of R^1;
    reconsider r = s as Real;
    assume
A5: s = w;
    thus L[01](t1,t2).s = U(r) by A1,Th7
      .= f.w by A3,A5;
  end;
A6: [.0,1.] = the carrier of Closed-Interval-TSpace(0,1) by TOPMETR:18;
A7: f is continuous by A3,TOPMETR:21;
  for s being Point of Closed-Interval-TSpace(0,1) holds L[01](t1,t2)
  is_continuous_at s
  proof
    let s be Point of Closed-Interval-TSpace(0,1);
    reconsider w = s as Point of R^1 by A6,TARSKI:def 3,TOPMETR:17;
    for G being Subset of Closed-Interval-TSpace(a,b) st G is open & L[01]
(t1,t2).s in G ex H being Subset of Closed-Interval-TSpace(0,1) st H is open &
    s in H & L[01](t1,t2).:H c= G
    proof
      let G be Subset of Closed-Interval-TSpace(a,b);
      assume G is open;
      then consider G0 being Subset of R^1 such that
A8:   G0 is open and
A9:   G0 /\ [#] Closed-Interval-TSpace(a,b) = G by TOPS_2:24;
A10:   f is_continuous_at w by A7,TMAP_1:44;
      assume L[01](t1,t2).s in G;
      then f.w in G by A4;
      then f.w in G0 by A9,XBOOLE_0:def 4;
      then consider H0 being Subset of R^1 such that
A11:  H0 is open and
A12:  w in H0 and
A13:  f.:H0 c= G0 by A8,A10,TMAP_1:43;
      now
        reconsider H = H0 /\ [#] Closed-Interval-TSpace(0,1) as Subset of
        Closed-Interval-TSpace(0,1);
        take H;
        thus H is open by A11,TOPS_2:24;
        thus s in H by A12,XBOOLE_0:def 4;
        thus L[01](t1,t2).:H c= G
        proof
          let t be object;
          assume t in L[01](t1,t2).:H;
          then consider r be object such that
          r in dom L[01](t1,t2) and
A14:      r in H and
A15:      t = L[01](t1,t2).r by FUNCT_1:def 6;
A16:      r in the carrier of Closed-Interval-TSpace(0,1) by A14;
          reconsider r as Point of Closed-Interval-TSpace(0,1) by A14;
          r in dom L[01](t1,t2) by A16,FUNCT_2:def 1;
          then
A17:      t in L[01](t1,t2).:(the carrier of Closed-Interval-TSpace(0,1))
          by A15,FUNCT_1:def 6;
          reconsider p = r as Point of R^1 by A6,TARSKI:def 3,TOPMETR:17;
          p in [#] R^1;
          then
A18:      p in dom f by FUNCT_2:def 1;
          t=f.p & p in H0 by A4,A14,A15,XBOOLE_0:def 4;
          then t in f.:H0 by A18,FUNCT_1:def 6;
          hence thesis by A9,A13,A17,XBOOLE_0:def 4;
        end;
      end;
      hence thesis;
    end;
    hence thesis by TMAP_1:43;
  end;
  hence thesis by TMAP_1:44;
end;
