Article HOMOTHET, MML version 4.99.1005

:: HOMOTHET:th 1
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2, b3, b4, b5, b6, b7, b8, b9, b10 being Element of the carrier of b1
      st not LIN b2,b3,b4 & LIN b2,b3,b5 & LIN b2,b3,b6 & LIN b2,b3,b7 & LIN b2,b4,b8 & LIN b2,b4,b9 & LIN b2,b4,b10 & b4 <> b9 & b3 <> b6 & b2 <> b9 & b2 <> b6 & b3,b4 // b5,b8 & b3,b9 // b5,b10 & b6,b4 // b7,b8 & b1 is Desarguesian
   holds b6,b9 // b7,b10;

:: HOMOTHET:th 2
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
      st for b2, b3, b4 being Element of the carrier of b1
              st b2 <> b3 & b2 <> b4 & LIN b2,b3,b4
           holds ex b5 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1 st
              b5 is dilatation(b1) & b5 . b2 = b2 & b5 . b3 = b4
   holds b1 is Desarguesian;

:: HOMOTHET:th 3
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
   st b1 is Desarguesian
for b2, b3, b4 being Element of the carrier of b1
      st b2 <> b3 & b2 <> b4 & LIN b2,b3,b4
   holds ex b5 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1 st
      b5 is dilatation(b1) & b5 . b2 = b2 & b5 . b3 = b4;

:: HOMOTHET:th 4
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
      b1 is Desarguesian
   iff
      for b2, b3, b4 being Element of the carrier of b1
            st b2 <> b3 & b2 <> b4 & LIN b2,b3,b4
         holds ex b5 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1 st
            b5 is dilatation(b1) & b5 . b2 = b2 & b5 . b3 = b4;

:: HOMOTHET:prednot 1 => HOMOTHET:pred 1
definition
  let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
  let a2 be Function-like quasi_total bijective Relation of the carrier of a1,the carrier of a1;
  let a3 be Element of bool the carrier of a1;
  pred A2 is_Sc A3 means
    a2 is collineation(a1) &
     a3 is being_line(a1) &
     (for b1 being Element of the carrier of a1
           st b1 in a3
        holds a2 . b1 = b1) &
     (for b1 being Element of the carrier of a1 holds
        b1,a2 . b1 // a3);
end;

:: HOMOTHET:dfs 1
definiens
  let a1 be non empty non trivial AffinSpace-like 2-dimensional AffinStruct;
  let a2 be Function-like quasi_total bijective Relation of the carrier of a1,the carrier of a1;
  let a3 be Element of bool the carrier of a1;
To prove
     a2 is_Sc a3
it is sufficient to prove
  thus a2 is collineation(a1) &
     a3 is being_line(a1) &
     (for b1 being Element of the carrier of a1
           st b1 in a3
        holds a2 . b1 = b1) &
     (for b1 being Element of the carrier of a1 holds
        b1,a2 . b1 // a3);

:: HOMOTHET:def 1
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
for b3 being Element of bool the carrier of b1 holds
      b2 is_Sc b3
   iff
      b2 is collineation(b1) &
       b3 is being_line(b1) &
       (for b4 being Element of the carrier of b1
             st b4 in b3
          holds b2 . b4 = b4) &
       (for b4 being Element of the carrier of b1 holds
          b4,b2 . b4 // b3);

:: HOMOTHET:th 5
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1
for b4 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1
      st b4 is_Sc b3 & b4 . b2 = b2 & not b2 in b3
   holds b4 = id the carrier of b1;

:: HOMOTHET:th 6
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
      st for b2, b3 being Element of the carrier of b1
        for b4 being Element of bool the carrier of b1
              st b2,b3 // b4 & not b2 in b4
           holds ex b5 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1 st
              b5 is_Sc b4 & b5 . b2 = b3
   holds b1 is Moufangian;

:: HOMOTHET:th 7
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2, b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of b1
for b10, b11 being Element of bool the carrier of b1
      st b10 // b11 & b2 in b10 & b3 in b10 & b4 in b10 & b5 in b10 & b1 is Moufangian & b6 in b11 & b7 in b11 & b8 in b11 & b9 in b11 & b6 <> b7 & b3 <> b2 & b2,b6 // b4,b8 & b2,b7 // b4,b9 & b3,b6 // b5,b8
   holds b3,b7 // b5,b9;

:: HOMOTHET:th 8
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct
for b2, b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of b1
      st b2,b3 // b4 & not b2 in b4 & b1 is Moufangian
   holds ex b5 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1 st
      b5 is_Sc b4 & b5 . b2 = b3;

:: HOMOTHET:th 9
theorem
for b1 being non empty non trivial AffinSpace-like 2-dimensional AffinStruct holds
      b1 is Moufangian
   iff
      for b2, b3 being Element of the carrier of b1
      for b4 being Element of bool the carrier of b1
            st b2,b3 // b4 & not b2 in b4
         holds ex b5 being Function-like quasi_total bijective Relation of the carrier of b1,the carrier of b1 st
            b5 is_Sc b4 & b5 . b2 = b3;