reserve T,T1,T2,S for non empty TopSpace;
reserve p,q for Point of TOP-REAL 2;

theorem Th43:
  for A,B,C,D being Real, f being Function of TOP-REAL 2,
TOP-REAL 2 st (for t being Point of TOP-REAL 2 holds f.t=|[A*(t`1)+B,C*(t`2)+D
  ]|) holds f is continuous
proof
  reconsider h11=proj1 as Function of TOP-REAL 2,R^1 by TOPMETR:17;
  set K0=[#](TOP-REAL 2);
  let A,B,C,D be Real,f be Function of TOP-REAL 2,TOP-REAL 2;
A1: (TOP-REAL 2)| [#](TOP-REAL 2)=the TopStruct of TOP-REAL 2 by TSEP_1:93;
  then reconsider h1=h11 as Function of (TOP-REAL 2)| [#](TOP-REAL 2),R^1;
  h11 is continuous by JORDAN5A:27;
  then h1 is continuous by A1,PRE_TOPC:32;
  then consider
  g1 being Function of (TOP-REAL 2)| [#](TOP-REAL 2),R^1 such that
A2: for p being Point of (TOP-REAL 2)| [#](TOP-REAL 2),r1 being Real
 st h1.p=r1 holds g1.p=A*r1 and
A3: g1 is continuous by Th23;
  reconsider f1=proj1*f as Function of (TOP-REAL 2)| [#](TOP-REAL 2),R^1 by A1,
TOPMETR:17;
  consider g11 being Function of (TOP-REAL 2)| [#](TOP-REAL 2),R^1 such that
A4: for p being Point of (TOP-REAL 2)| [#](TOP-REAL 2),r1 being Real
 st g1.p=r1 holds g11.p=r1+B and
A5: g11 is continuous by A3,Th24;
  reconsider f2=proj2*f as Function of (TOP-REAL 2)| [#](TOP-REAL 2),R^1 by A1,
TOPMETR:17;
  reconsider h11=proj2 as Function of TOP-REAL 2,R^1 by TOPMETR:17;
  reconsider h1=h11 as Function of (TOP-REAL 2)| [#](TOP-REAL 2),R^1 by A1;
  dom f1=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
  then
A6: dom f1=dom g11 by A1,FUNCT_2:def 1;
  assume
A7: for t being Point of TOP-REAL 2 holds f.t=|[A*(t`1)+B,C*(t`2)+D]|;
A8: for x being object st x in dom f1 holds f1.x=g11.x
  proof
    let x be object;
    assume
A9: x in dom f1;
    then reconsider p=x as Point of TOP-REAL 2 by FUNCT_2:def 1;
    f1.x=proj1.(f.x) by A9,FUNCT_1:12;
    then
A10: f1.x=proj1.(|[A*(p`1)+B,C*(p`2)+D]|) by A7
      .=A*(p`1)+B by PSCOMP_1:65
      .=A*(proj1.p)+B by PSCOMP_1:def 5;
    A*(proj1.p)=g1.p by A1,A2;
    hence thesis by A1,A4,A10;
  end;
  h11 is continuous by JORDAN5A:27;
  then h1 is continuous by A1,PRE_TOPC:32;
  then consider
  g1 being Function of (TOP-REAL 2)| [#](TOP-REAL 2),R^1 such that
A11: for p being Point of (TOP-REAL 2)| [#](TOP-REAL 2),r1 being Real
 st h1.p=r1 holds g1.p=C*r1 and
A12: g1 is continuous by Th23;
  consider g11 being Function of (TOP-REAL 2)| [#](TOP-REAL 2),R^1 such that
A13: for p being Point of (TOP-REAL 2)| [#](TOP-REAL 2),r1 being Real
 st g1.p=r1 holds g11.p=r1+D and
A14: g11 is continuous by A12,Th24;
A15: for x being object st x in dom f2 holds f2.x=g11.x
  proof
    let x be object;
    assume
A16: x in dom f2;
    then reconsider p=x as Point of TOP-REAL 2 by FUNCT_2:def 1;
    f2.x=proj2.(f.x) by A16,FUNCT_1:12;
    then
A17: f2.x=proj2.(|[A*(p`1)+B,C*(p`2)+D]|) by A7
      .=C*(p`2)+D by PSCOMP_1:65
      .=C*(proj2.p)+D by PSCOMP_1:def 6;
    C*(proj2.p)=g1.p by A1,A11;
    hence thesis by A1,A13,A17;
  end;
  reconsider f0=f as Function of (TOP-REAL 2)| [#](TOP-REAL 2), (TOP-REAL 2)|
  [#](TOP-REAL 2) by A1;
A18: for x,y,r,s being Real st |[x,y]| in K0 & r=f1.(|[x,y]|) & s=f2.
  (|[x,y]|) holds f0. |[x,y]|=|[r,s]|
  proof
    let x,y,r,s be Real;
    assume that
    |[x,y]| in K0 and
A19: r=f1.(|[x,y]|) & s=f2.(|[x,y]|);
A20: f. |[x,y]| is Point of TOP-REAL 2;
    dom f =the carrier of TOP-REAL 2 by FUNCT_2:def 1;
    then proj1.(f0. |[x,y]|) =r & proj2.(f0. |[x,y]|) =s by A19,FUNCT_1:13;
    hence thesis by A20,Th8;
  end;
  dom f2=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
  then dom f2=dom g11 by A1,FUNCT_2:def 1;
  then
A21: f2 is continuous by A14,A15,FUNCT_1:2;
  f1 is continuous by A5,A6,A8,FUNCT_1:2;
  then f0 is continuous by A21,A18,Th35;
  hence f is continuous by A1,PRE_TOPC:34;
end;
