reserve a,b,c,d for Real;

theorem Th12:
  a < b implies for t1,t2 being Point of Closed-Interval-TSpace(0,1)
  holds P[01](a,b,t1,t2) is continuous
proof
  assume
A1: a < b;
  reconsider a, b as Real;
A2: [.a,b.] = the carrier of Closed-Interval-TSpace(a,b) by A1,TOPMETR:18;
  let t1,t2 be Point of Closed-Interval-TSpace(0,1);
  reconsider r1 = t1, r2 = t2 as Real;
  deffunc U(Real) = In(((r2 - r1)/(b-a))*$1 + (b*r1 -a*r2)/(b-a),REAL);
  consider P being Function of REAL,REAL such that
A3: for r being Element of REAL holds P.r= U(r) from FUNCT_2:sch 4;
A4: for r being Real
    holds P.r= ((r2 - r1)/(b-a))*r + (b*r1 -a*r2)/(b-a)
   proof let r be Real;
     reconsider r as Element of REAL by XREAL_0:def 1;
     P.r= U(r) by A3;
    hence thesis;
   end;
  reconsider f = P as Function of R^1, R^1 by TOPMETR:17;
A5: for s being Point of Closed-Interval-TSpace(a,b), w being Point of R^1
  st s = w holds P[01](a,b,t1,t2).s = f.w
  proof
    let s be Point of Closed-Interval-TSpace(a,b), w be Point of R^1;
    reconsider r = s as Real;
    assume
A6: s = w;
    thus P[01](a,b,t1,t2).s = ((r2 - r1)/(b-a))*r + (b*r1 -a*r2)/(b-a) by A1
,Th11
      .= U(r)
      .= f.w by A4,A6;
  end;
A7: f is continuous by A4,TOPMETR:21;
  for s being Point of Closed-Interval-TSpace(a,b) holds P[01](a,b,t1,t2)
  is_continuous_at s
  proof
    let s be Point of Closed-Interval-TSpace(a,b);
    reconsider w = s as Point of R^1 by A2,TARSKI:def 3,TOPMETR:17;
    for G being Subset of Closed-Interval-TSpace(0,1) st G is open & P[01]
    (a,b,t1,t2).s in G ex H being Subset of Closed-Interval-TSpace(a,b) st H is
    open & s in H & P[01](a,b,t1,t2).:H c= G
    proof
      let G be Subset of Closed-Interval-TSpace(0,1);
      assume G is open;
      then consider G0 being Subset of R^1 such that
A8:   G0 is open and
A9:   G0 /\ [#] Closed-Interval-TSpace(0,1) = G by TOPS_2:24;
A10:   f is_continuous_at w by A7,TMAP_1:44;
      assume P[01](a,b,t1,t2).s in G;
      then f.w in G by A5;
      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(a,b) as Subset of
        Closed-Interval-TSpace(a,b);
        take H;
        thus H is open by A11,TOPS_2:24;
        thus s in H by A12,XBOOLE_0:def 4;
        thus P[01](a,b,t1,t2).:H c= G
        proof
          let t be object;
          assume t in P[01](a,b,t1,t2).:H;
          then consider r be object such that
          r in dom P[01](a,b,t1,t2) and
A14:      r in H and
A15:      t = P[01](a,b,t1,t2).r by FUNCT_1:def 6;
A16:      r in the carrier of Closed-Interval-TSpace(a,b) by A14;
          reconsider r as Point of Closed-Interval-TSpace(a,b) by A14;
          r in dom P[01](a,b,t1,t2) by A16,FUNCT_2:def 1;
          then
A17:      t in P[01](a,b,t1,t2).: (the carrier of Closed-Interval-TSpace(
          a,b)) by A15,FUNCT_1:def 6;
          reconsider p = r as Point of R^1 by A2,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 A5,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;
