reserve n for Element of NAT,
  a, b for Real;

theorem Th6:
  for F being Function of [:R^1,I[01]:], R^1 st for x being Point
of R^1, i being Point of I[01] holds F.(x,i) = (1-i) * x holds F is continuous
proof
  deffunc Fa(Element of TOP-REAL 1, Element of I) = (1-$2)*$1;
  consider G being Function of [:the carrier of TOP-REAL 1,I:], the carrier of
  TOP-REAL 1 such that
A1: for x being Point of TOP-REAL 1, i being Point of I[01] holds G.(x,i
  ) = Fa(x,i) from BINOP_1:sch 4;
  reconsider G as Function of [:TOP-REAL 1,I[01]:], TOP-REAL 1 by Lm2;
  consider f being Function of TOP-REAL 1,R^1 such that
A2: for p being Element of TOP-REAL 1 holds f.p = p/.1 by JORDAN2B:1;
A3: f is being_homeomorphism by A2,JORDAN2B:28;
  then
A4: f is continuous by TOPS_2:def 5;
  let F be Function of [:R^1,I[01]:], R^1 such that
A5: for x being Point of R^1, i being Point of I[01] holds F.(x,i) = (1-
  i) * x;
A6: for x being Point of [:R^1,I[01]:] holds F.x = (f*(G*[:f",id I[01]:])).x
  proof
    reconsider ff = f as Function;
    let x be Point of [:R^1,I[01]:];
    consider a being Point of R^1, b being Point of I[01] such that
A7: x = [a,b] by BORSUK_1:10;
A8: dom (f") = the carrier of R^1 by FUNCT_2:def 1;
 rng f = [#]R^1 by A3,TOPS_2:def 5;
    then
A9: f is onto by FUNCT_2:def 3;
A10: dom f = the carrier of TOP-REAL 1 by FUNCT_2:def 1;
    set g = ff";
    consider z being Real such that
A11: (1-b)*f".a = <*z*> by JORDAN2B:20;
    reconsider zz=z as Element of REAL by XREAL_0:def 1;
A12: <*a*> = |[a]|;
    then reconsider w = <*a*> as Element of REAL 1 by EUCLID:22;
A13: ff is one-to-one by A3,TOPS_2:def 5;
    f.<*a*> = |[a]|/.1 by A2
      .= a by Th4;
    then <*a*> = g.a by A10,A13,A12,FUNCT_1:32;
    then
A14: w = f/".a by A9,A13,TOPS_2:def 4;
A15: <*z*> = (1-b)*(f/".a) by A11
      .= (1-b)*w by A14
      .= <*(1-b)*a*> by RVSUM_1:47;
    thus (f*(G*[:f",id I[01]:])).x = f.((G*[:f",id I[01]:]).x) by FUNCT_2:15
      .= f.(G.([:f",id I[01]:].(a,b))) by A7,FUNCT_2:15
      .= f.(G.(f".a,id I[01].b)) by A8,Lm4,FUNCT_3:def 8
      .= f.(G.(f".a,b)) by FUNCT_1:18
      .= f.((1-b)*f".a) by A1
      .= ((1-b)*f".a)/.1 by A2
      .= <*zz*>/.1 by A11
      .= z by FINSEQ_4:16
      .= (1-b)*a by A15,FINSEQ_1:76
      .= F.(a,b) by A5
      .= F.x by A7;
  end;
  f" is continuous by A3,TOPS_2:def 5;
  then
A16: [:f",id I[01]:] is continuous;
  G is continuous by A1,TOPALG_1:17;
  hence thesis by A4,A16,A6,FUNCT_2:63;
end;
