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

theorem Th44:
  for A,B,C,D being Real st A>0 & C>0 holds AffineMap(A,B,C
  ,D) is one-to-one
proof
  let A,B,C,D be Real such that
A1: A>0 and
A2: C>0;
  set ff = AffineMap(A,B,C,D);
  for x1,x2 being object st x1 in dom ff & x2 in dom ff & ff.x1=ff.x2
holds x1=x2
  proof
    let x1,x2 be object;
    assume that
A3: x1 in dom ff and
A4: x2 in dom ff and
A5: ff.x1=ff.x2;
    reconsider p2=x2 as Point of TOP-REAL 2 by A4;
    reconsider p1=x1 as Point of TOP-REAL 2 by A3;
A6: ff.x1= |[A*(p1`1)+B,C*(p1`2)+D]| & ff.x2= |[A*(p2`1)+B,C*(p2`2)+D]| by Def2
;
    then A*(p1`1)+B=A*(p2`1)+B by A5,SPPOL_2:1;
    then (p1`1)=A*(p2`1)/A by A1,XCMPLX_1:89;
    then
A7: (p1`1)=(p2`1) by A1,XCMPLX_1:89;
    C*(p1`2)+D=C*(p2`2)+D by A5,A6,SPPOL_2:1;
    then (p1`2)=C*(p2`2)/C by A2,XCMPLX_1:89;
    hence thesis by A2,A7,TOPREAL3:6,XCMPLX_1:89;
  end;
  hence thesis by FUNCT_1:def 4;
end;
