Article INCSP_1, MML version 4.99.1005

:: INCSP_1:structnot 1 => INCSP_1:struct 1
definition
  struct() IncProjStr(#
    Points -> non empty set,
    Lines -> non empty set,
    Inc -> Relation of the Points of it,the Lines of it
  #);
end;

:: INCSP_1:attrnot 1 => INCSP_1:attr 1
definition
  let a1 be IncProjStr;
  attr a1 is strict;
end;

:: INCSP_1:exreg 1
registration
  cluster strict IncProjStr;
end;

:: INCSP_1:aggrnot 1 => INCSP_1:aggr 1
definition
  let a1, a2 be non empty set;
  let a3 be Relation of a1,a2;
  aggr IncProjStr(#a1,a2,a3#) -> strict IncProjStr;
end;

:: INCSP_1:selnot 1 => INCSP_1:sel 1
definition
  let a1 be IncProjStr;
  sel the Points of a1 -> non empty set;
end;

:: INCSP_1:selnot 2 => INCSP_1:sel 2
definition
  let a1 be IncProjStr;
  sel the Lines of a1 -> non empty set;
end;

:: INCSP_1:selnot 3 => INCSP_1:sel 3
definition
  let a1 be IncProjStr;
  sel the Inc of a1 -> Relation of the Points of a1,the Lines of a1;
end;

:: INCSP_1:structnot 2 => INCSP_1:struct 2
definition
  struct(IncProjStr) IncStruct(#
    Points -> non empty set,
    Lines -> non empty set,
    Planes -> non empty set,
    Inc -> Relation of the Points of it,the Lines of it,
    Inc2 -> Relation of the Points of it,the Planes of it,
    Inc3 -> Relation of the Lines of it,the Planes of it
  #);
end;

:: INCSP_1:attrnot 2 => INCSP_1:attr 2
definition
  let a1 be IncStruct;
  attr a1 is strict;
end;

:: INCSP_1:exreg 2
registration
  cluster strict IncStruct;
end;

:: INCSP_1:aggrnot 2 => INCSP_1:aggr 2
definition
  let a1, a2, a3 be non empty set;
  let a4 be Relation of a1,a2;
  let a5 be Relation of a1,a3;
  let a6 be Relation of a2,a3;
  aggr IncStruct(#a1,a2,a3,a4,a5,a6#) -> strict IncStruct;
end;

:: INCSP_1:selnot 4 => INCSP_1:sel 4
definition
  let a1 be IncStruct;
  sel the Planes of a1 -> non empty set;
end;

:: INCSP_1:selnot 5 => INCSP_1:sel 5
definition
  let a1 be IncStruct;
  sel the Inc2 of a1 -> Relation of the Points of a1,the Planes of a1;
end;

:: INCSP_1:selnot 6 => INCSP_1:sel 6
definition
  let a1 be IncStruct;
  sel the Inc3 of a1 -> Relation of the Lines of a1,the Planes of a1;
end;

:: INCSP_1:modenot 1
definition
  let a1 be IncProjStr;
  mode POINT of a1 is Element of the Points of a1;
end;

:: INCSP_1:modenot 2
definition
  let a1 be IncProjStr;
  mode LINE of a1 is Element of the Lines of a1;
end;

:: INCSP_1:modenot 3
definition
  let a1 be IncStruct;
  mode PLANE of a1 is Element of the Planes of a1;
end;

:: INCSP_1:prednot 1 => INCSP_1:pred 1
definition
  let a1 be IncProjStr;
  let a2 be Element of the Points of a1;
  let a3 be Element of the Lines of a1;
  pred A2 on A3 means
    [a2,a3] in the Inc of a1;
end;

:: INCSP_1:dfs 1
definiens
  let a1 be IncProjStr;
  let a2 be Element of the Points of a1;
  let a3 be Element of the Lines of a1;
To prove
     a2 on a3
it is sufficient to prove
  thus [a2,a3] in the Inc of a1;

:: INCSP_1:def 1
theorem
for b1 being IncProjStr
for b2 being Element of the Points of b1
for b3 being Element of the Lines of b1 holds
      b2 on b3
   iff
      [b2,b3] in the Inc of b1;

:: INCSP_1:prednot 2 => INCSP_1:pred 2
definition
  let a1 be IncStruct;
  let a2 be Element of the Points of a1;
  let a3 be Element of the Planes of a1;
  pred A2 on A3 means
    [a2,a3] in the Inc2 of a1;
end;

:: INCSP_1:dfs 2
definiens
  let a1 be IncStruct;
  let a2 be Element of the Points of a1;
  let a3 be Element of the Planes of a1;
To prove
     a2 on a3
it is sufficient to prove
  thus [a2,a3] in the Inc2 of a1;

:: INCSP_1:def 2
theorem
for b1 being IncStruct
for b2 being Element of the Points of b1
for b3 being Element of the Planes of b1 holds
      b2 on b3
   iff
      [b2,b3] in the Inc2 of b1;

:: INCSP_1:prednot 3 => INCSP_1:pred 3
definition
  let a1 be IncStruct;
  let a2 be Element of the Lines of a1;
  let a3 be Element of the Planes of a1;
  pred A2 on A3 means
    [a2,a3] in the Inc3 of a1;
end;

:: INCSP_1:dfs 3
definiens
  let a1 be IncStruct;
  let a2 be Element of the Lines of a1;
  let a3 be Element of the Planes of a1;
To prove
     a2 on a3
it is sufficient to prove
  thus [a2,a3] in the Inc3 of a1;

:: INCSP_1:def 3
theorem
for b1 being IncStruct
for b2 being Element of the Lines of b1
for b3 being Element of the Planes of b1 holds
      b2 on b3
   iff
      [b2,b3] in the Inc3 of b1;

:: INCSP_1:prednot 4 => INCSP_1:pred 4
definition
  let a1 be IncProjStr;
  let a2 be Element of bool the Points of a1;
  let a3 be Element of the Lines of a1;
  pred A2 on A3 means
    for b1 being Element of the Points of a1
          st b1 in a2
       holds b1 on a3;
end;

:: INCSP_1:dfs 4
definiens
  let a1 be IncProjStr;
  let a2 be Element of bool the Points of a1;
  let a3 be Element of the Lines of a1;
To prove
     a2 on a3
it is sufficient to prove
  thus for b1 being Element of the Points of a1
          st b1 in a2
       holds b1 on a3;

:: INCSP_1:def 4
theorem
for b1 being IncProjStr
for b2 being Element of bool the Points of b1
for b3 being Element of the Lines of b1 holds
      b2 on b3
   iff
      for b4 being Element of the Points of b1
            st b4 in b2
         holds b4 on b3;

:: INCSP_1:prednot 5 => INCSP_1:pred 5
definition
  let a1 be IncStruct;
  let a2 be Element of bool the Points of a1;
  let a3 be Element of the Planes of a1;
  pred A2 on A3 means
    for b1 being Element of the Points of a1
          st b1 in a2
       holds b1 on a3;
end;

:: INCSP_1:dfs 5
definiens
  let a1 be IncStruct;
  let a2 be Element of bool the Points of a1;
  let a3 be Element of the Planes of a1;
To prove
     a2 on a3
it is sufficient to prove
  thus for b1 being Element of the Points of a1
          st b1 in a2
       holds b1 on a3;

:: INCSP_1:def 5
theorem
for b1 being IncStruct
for b2 being Element of bool the Points of b1
for b3 being Element of the Planes of b1 holds
      b2 on b3
   iff
      for b4 being Element of the Points of b1
            st b4 in b2
         holds b4 on b3;

:: INCSP_1:attrnot 3 => INCSP_1:attr 3
definition
  let a1 be IncProjStr;
  let a2 be Element of bool the Points of a1;
  attr a2 is linear means
    ex b1 being Element of the Lines of a1 st
       a2 on b1;
end;

:: INCSP_1:dfs 6
definiens
  let a1 be IncProjStr;
  let a2 be Element of bool the Points of a1;
To prove
     a2 is linear
it is sufficient to prove
  thus ex b1 being Element of the Lines of a1 st
       a2 on b1;

:: INCSP_1:def 6
theorem
for b1 being IncProjStr
for b2 being Element of bool the Points of b1 holds
      b2 is linear(b1)
   iff
      ex b3 being Element of the Lines of b1 st
         b2 on b3;

:: INCSP_1:prednot 6 => INCSP_1:attr 3
notation
  let a1 be IncProjStr;
  let a2 be Element of bool the Points of a1;
  synonym a2 is_collinear for linear;
end;

:: INCSP_1:attrnot 4 => INCSP_1:attr 4
definition
  let a1 be IncStruct;
  let a2 be Element of bool the Points of a1;
  attr a2 is planar means
    ex b1 being Element of the Planes of a1 st
       a2 on b1;
end;

:: INCSP_1:dfs 7
definiens
  let a1 be IncStruct;
  let a2 be Element of bool the Points of a1;
To prove
     a2 is planar
it is sufficient to prove
  thus ex b1 being Element of the Planes of a1 st
       a2 on b1;

:: INCSP_1:def 7
theorem
for b1 being IncStruct
for b2 being Element of bool the Points of b1 holds
      b2 is planar(b1)
   iff
      ex b3 being Element of the Planes of b1 st
         b2 on b3;

:: INCSP_1:prednot 7 => INCSP_1:attr 4
notation
  let a1 be IncStruct;
  let a2 be Element of bool the Points of a1;
  synonym a2 is_coplanar for planar;
end;

:: INCSP_1:th 11
theorem
for b1 being IncProjStr
for b2 being Element of the Lines of b1
for b3, b4 being Element of the Points of b1 holds
   {b3,b4} on b2
iff
   b3 on b2 & b4 on b2;

:: INCSP_1:th 12
theorem
for b1 being IncProjStr
for b2 being Element of the Lines of b1
for b3, b4, b5 being Element of the Points of b1 holds
   {b3,b4,b5} on b2
iff
   b3 on b2 & b4 on b2 & b5 on b2;

:: INCSP_1:th 13
theorem
for b1 being IncStruct
for b2, b3 being Element of the Points of b1
for b4 being Element of the Planes of b1 holds
      {b2,b3} on b4
   iff
      b2 on b4 & b3 on b4;

:: INCSP_1:th 14
theorem
for b1 being IncStruct
for b2, b3, b4 being Element of the Points of b1
for b5 being Element of the Planes of b1 holds
      {b2,b3,b4} on b5
   iff
      b2 on b5 & b3 on b5 & b4 on b5;

:: INCSP_1:th 15
theorem
for b1 being IncStruct
for b2, b3, b4, b5 being Element of the Points of b1
for b6 being Element of the Planes of b1 holds
      {b2,b3,b4,b5} on b6
   iff
      b2 on b6 & b3 on b6 & b4 on b6 & b5 on b6;

:: INCSP_1:th 16
theorem
for b1 being IncStruct
for b2 being Element of the Lines of b1
for b3, b4 being Element of bool the Points of b1
      st b3 c= b4 & b4 on b2
   holds b3 on b2;

:: INCSP_1:th 17
theorem
for b1 being IncStruct
for b2 being Element of the Planes of b1
for b3, b4 being Element of bool the Points of b1
      st b3 c= b4 & b4 on b2
   holds b3 on b2;

:: INCSP_1:th 18
theorem
for b1 being IncStruct
for b2 being Element of the Points of b1
for b3 being Element of the Lines of b1
for b4 being Element of bool the Points of b1 holds
      b4 on b3 & b2 on b3
   iff
      b4 \/ {b2} on b3;

:: INCSP_1:th 19
theorem
for b1 being IncStruct
for b2 being Element of the Points of b1
for b3 being Element of the Planes of b1
for b4 being Element of bool the Points of b1 holds
      b4 on b3 & b2 on b3
   iff
      b4 \/ {b2} on b3;

:: INCSP_1:th 20
theorem
for b1 being IncStruct
for b2 being Element of the Lines of b1
for b3, b4 being Element of bool the Points of b1 holds
   b3 \/ b4 on b2
iff
   b3 on b2 & b4 on b2;

:: INCSP_1:th 21
theorem
for b1 being IncStruct
for b2 being Element of the Planes of b1
for b3, b4 being Element of bool the Points of b1 holds
   b3 \/ b4 on b2
iff
   b3 on b2 & b4 on b2;

:: INCSP_1:th 22
theorem
for b1 being IncStruct
for b2, b3 being Element of bool the Points of b1
      st b2 c= b3 & b3 is linear(b1)
   holds b2 is linear(b1);

:: INCSP_1:th 23
theorem
for b1 being IncStruct
for b2, b3 being Element of bool the Points of b1
      st b2 c= b3 & b3 is planar(b1)
   holds b2 is planar(b1);

:: INCSP_1:attrnot 5 => INCSP_1:attr 5
definition
  let a1 be IncProjStr;
  attr a1 is with_non-trivial_lines means
    for b1 being Element of the Lines of a1 holds
       ex b2, b3 being Element of the Points of a1 st
          b2 <> b3 & {b2,b3} on b1;
end;

:: INCSP_1:dfs 8
definiens
  let a1 be IncProjStr;
To prove
     a1 is with_non-trivial_lines
it is sufficient to prove
  thus for b1 being Element of the Lines of a1 holds
       ex b2, b3 being Element of the Points of a1 st
          b2 <> b3 & {b2,b3} on b1;

:: INCSP_1:def 8
theorem
for b1 being IncProjStr holds
      b1 is with_non-trivial_lines
   iff
      for b2 being Element of the Lines of b1 holds
         ex b3, b4 being Element of the Points of b1 st
            b3 <> b4 & {b3,b4} on b2;

:: INCSP_1:attrnot 6 => INCSP_1:attr 6
definition
  let a1 be IncProjStr;
  attr a1 is linear means
    for b1, b2 being Element of the Points of a1 holds
    ex b3 being Element of the Lines of a1 st
       {b1,b2} on b3;
end;

:: INCSP_1:dfs 9
definiens
  let a1 be IncProjStr;
To prove
     a1 is linear
it is sufficient to prove
  thus for b1, b2 being Element of the Points of a1 holds
    ex b3 being Element of the Lines of a1 st
       {b1,b2} on b3;

:: INCSP_1:def 9
theorem
for b1 being IncProjStr holds
      b1 is linear
   iff
      for b2, b3 being Element of the Points of b1 holds
      ex b4 being Element of the Lines of b1 st
         {b2,b3} on b4;

:: INCSP_1:attrnot 7 => INCSP_1:attr 7
definition
  let a1 be IncProjStr;
  attr a1 is up-2-rank means
    for b1, b2 being Element of the Points of a1
    for b3, b4 being Element of the Lines of a1
          st b1 <> b2 & {b1,b2} on b3 & {b1,b2} on b4
       holds b3 = b4;
end;

:: INCSP_1:dfs 10
definiens
  let a1 be IncProjStr;
To prove
     a1 is up-2-rank
it is sufficient to prove
  thus for b1, b2 being Element of the Points of a1
    for b3, b4 being Element of the Lines of a1
          st b1 <> b2 & {b1,b2} on b3 & {b1,b2} on b4
       holds b3 = b4;

:: INCSP_1:def 10
theorem
for b1 being IncProjStr holds
      b1 is up-2-rank
   iff
      for b2, b3 being Element of the Points of b1
      for b4, b5 being Element of the Lines of b1
            st b2 <> b3 & {b2,b3} on b4 & {b2,b3} on b5
         holds b4 = b5;

:: INCSP_1:attrnot 8 => INCSP_1:attr 8
definition
  let a1 be IncStruct;
  attr a1 is with_non-empty_planes means
    for b1 being Element of the Planes of a1 holds
       ex b2 being Element of the Points of a1 st
          b2 on b1;
end;

:: INCSP_1:dfs 11
definiens
  let a1 be IncStruct;
To prove
     a1 is with_non-empty_planes
it is sufficient to prove
  thus for b1 being Element of the Planes of a1 holds
       ex b2 being Element of the Points of a1 st
          b2 on b1;

:: INCSP_1:def 11
theorem
for b1 being IncStruct holds
      b1 is with_non-empty_planes
   iff
      for b2 being Element of the Planes of b1 holds
         ex b3 being Element of the Points of b1 st
            b3 on b2;

:: INCSP_1:attrnot 9 => INCSP_1:attr 9
definition
  let a1 be IncStruct;
  attr a1 is planar means
    for b1, b2, b3 being Element of the Points of a1 holds
    ex b4 being Element of the Planes of a1 st
       {b1,b2,b3} on b4;
end;

:: INCSP_1:dfs 12
definiens
  let a1 be IncStruct;
To prove
     a1 is planar
it is sufficient to prove
  thus for b1, b2, b3 being Element of the Points of a1 holds
    ex b4 being Element of the Planes of a1 st
       {b1,b2,b3} on b4;

:: INCSP_1:def 12
theorem
for b1 being IncStruct holds
      b1 is planar
   iff
      for b2, b3, b4 being Element of the Points of b1 holds
      ex b5 being Element of the Planes of b1 st
         {b2,b3,b4} on b5;

:: INCSP_1:attrnot 10 => INCSP_1:attr 10
definition
  let a1 be IncStruct;
  attr a1 is with_<=1_plane_per_3_pts means
    for b1, b2, b3 being Element of the Points of a1
    for b4, b5 being Element of the Planes of a1
          st {b1,b2,b3} is not linear(a1) & {b1,b2,b3} on b4 & {b1,b2,b3} on b5
       holds b4 = b5;
end;

:: INCSP_1:dfs 13
definiens
  let a1 be IncStruct;
To prove
     a1 is with_<=1_plane_per_3_pts
it is sufficient to prove
  thus for b1, b2, b3 being Element of the Points of a1
    for b4, b5 being Element of the Planes of a1
          st {b1,b2,b3} is not linear(a1) & {b1,b2,b3} on b4 & {b1,b2,b3} on b5
       holds b4 = b5;

:: INCSP_1:def 13
theorem
for b1 being IncStruct holds
      b1 is with_<=1_plane_per_3_pts
   iff
      for b2, b3, b4 being Element of the Points of b1
      for b5, b6 being Element of the Planes of b1
            st {b2,b3,b4} is not linear(b1) & {b2,b3,b4} on b5 & {b2,b3,b4} on b6
         holds b5 = b6;

:: INCSP_1:attrnot 11 => INCSP_1:attr 11
definition
  let a1 be IncStruct;
  attr a1 is with_lines_inside_planes means
    for b1 being Element of the Lines of a1
    for b2 being Element of the Planes of a1
          st ex b3, b4 being Element of the Points of a1 st
               b3 <> b4 & {b3,b4} on b1 & {b3,b4} on b2
       holds b1 on b2;
end;

:: INCSP_1:dfs 14
definiens
  let a1 be IncStruct;
To prove
     a1 is with_lines_inside_planes
it is sufficient to prove
  thus for b1 being Element of the Lines of a1
    for b2 being Element of the Planes of a1
          st ex b3, b4 being Element of the Points of a1 st
               b3 <> b4 & {b3,b4} on b1 & {b3,b4} on b2
       holds b1 on b2;

:: INCSP_1:def 14
theorem
for b1 being IncStruct holds
      b1 is with_lines_inside_planes
   iff
      for b2 being Element of the Lines of b1
      for b3 being Element of the Planes of b1
            st ex b4, b5 being Element of the Points of b1 st
                 b4 <> b5 & {b4,b5} on b2 & {b4,b5} on b3
         holds b2 on b3;

:: INCSP_1:attrnot 12 => INCSP_1:attr 12
definition
  let a1 be IncStruct;
  attr a1 is with_planes_intersecting_in_2_pts means
    for b1 being Element of the Points of a1
    for b2, b3 being Element of the Planes of a1
          st b1 on b2 & b1 on b3
       holds ex b4 being Element of the Points of a1 st
          b1 <> b4 & b4 on b2 & b4 on b3;
end;

:: INCSP_1:dfs 15
definiens
  let a1 be IncStruct;
To prove
     a1 is with_planes_intersecting_in_2_pts
it is sufficient to prove
  thus for b1 being Element of the Points of a1
    for b2, b3 being Element of the Planes of a1
          st b1 on b2 & b1 on b3
       holds ex b4 being Element of the Points of a1 st
          b1 <> b4 & b4 on b2 & b4 on b3;

:: INCSP_1:def 15
theorem
for b1 being IncStruct holds
      b1 is with_planes_intersecting_in_2_pts
   iff
      for b2 being Element of the Points of b1
      for b3, b4 being Element of the Planes of b1
            st b2 on b3 & b2 on b4
         holds ex b5 being Element of the Points of b1 st
            b2 <> b5 & b5 on b3 & b5 on b4;

:: INCSP_1:attrnot 13 => INCSP_1:attr 13
definition
  let a1 be IncStruct;
  attr a1 is up-3-dimensional means
    ex b1, b2, b3, b4 being Element of the Points of a1 st
       {b1,b2,b3,b4} is not planar(a1);
end;

:: INCSP_1:dfs 16
definiens
  let a1 be IncStruct;
To prove
     a1 is up-3-dimensional
it is sufficient to prove
  thus ex b1, b2, b3, b4 being Element of the Points of a1 st
       {b1,b2,b3,b4} is not planar(a1);

:: INCSP_1:def 16
theorem
for b1 being IncStruct holds
      b1 is up-3-dimensional
   iff
      ex b2, b3, b4, b5 being Element of the Points of b1 st
         {b2,b3,b4,b5} is not planar(b1);

:: INCSP_1:attrnot 14 => INCSP_1:attr 14
definition
  let a1 be IncStruct;
  attr a1 is inc-compatible means
    for b1 being Element of the Points of a1
    for b2 being Element of the Lines of a1
    for b3 being Element of the Planes of a1
          st b1 on b2 & b2 on b3
       holds b1 on b3;
end;

:: INCSP_1:dfs 17
definiens
  let a1 be IncStruct;
To prove
     a1 is inc-compatible
it is sufficient to prove
  thus for b1 being Element of the Points of a1
    for b2 being Element of the Lines of a1
    for b3 being Element of the Planes of a1
          st b1 on b2 & b2 on b3
       holds b1 on b3;

:: INCSP_1:def 17
theorem
for b1 being IncStruct holds
      b1 is inc-compatible
   iff
      for b2 being Element of the Points of b1
      for b3 being Element of the Lines of b1
      for b4 being Element of the Planes of b1
            st b2 on b3 & b3 on b4
         holds b2 on b4;

:: INCSP_1:attrnot 15 => INCSP_1:attr 15
definition
  let a1 be IncStruct;
  attr a1 is IncSpace-like means
    a1 is with_non-trivial_lines & a1 is linear & a1 is up-2-rank & a1 is with_non-empty_planes & a1 is planar & a1 is with_<=1_plane_per_3_pts & a1 is with_lines_inside_planes & a1 is with_planes_intersecting_in_2_pts & a1 is up-3-dimensional & a1 is inc-compatible;
end;

:: INCSP_1:dfs 18
definiens
  let a1 be IncStruct;
To prove
     a1 is IncSpace-like
it is sufficient to prove
  thus a1 is with_non-trivial_lines & a1 is linear & a1 is up-2-rank & a1 is with_non-empty_planes & a1 is planar & a1 is with_<=1_plane_per_3_pts & a1 is with_lines_inside_planes & a1 is with_planes_intersecting_in_2_pts & a1 is up-3-dimensional & a1 is inc-compatible;

:: INCSP_1:def 18
theorem
for b1 being IncStruct holds
      b1 is IncSpace-like
   iff
      b1 is with_non-trivial_lines & b1 is linear & b1 is up-2-rank & b1 is with_non-empty_planes & b1 is planar & b1 is with_<=1_plane_per_3_pts & b1 is with_lines_inside_planes & b1 is with_planes_intersecting_in_2_pts & b1 is up-3-dimensional & b1 is inc-compatible;

:: INCSP_1:condreg 1
registration
  cluster IncSpace-like -> with_non-trivial_lines linear up-2-rank with_non-empty_planes planar with_<=1_plane_per_3_pts with_lines_inside_planes with_planes_intersecting_in_2_pts up-3-dimensional inc-compatible (IncStruct);
end;

:: INCSP_1:exreg 3
registration
  cluster strict IncSpace-like IncStruct;
end;

:: INCSP_1:modenot 4
definition
  mode IncSpace is IncSpace-like IncStruct;
end;

:: INCSP_1:th 35
theorem
for b1 being IncSpace-like IncStruct
for b2 being Element of the Lines of b1
for b3 being Element of the Planes of b1
for b4 being Element of bool the Points of b1
      st b4 on b2 & b2 on b3
   holds b4 on b3;

:: INCSP_1:th 36
theorem
for b1 being IncSpace-like IncStruct
for b2, b3 being Element of the Points of b1 holds
{b2,b2,b3} is linear(b1);

:: INCSP_1:th 37
theorem
for b1 being IncSpace-like IncStruct
for b2, b3, b4 being Element of the Points of b1 holds
{b2,b2,b3,b4} is planar(b1);

:: INCSP_1:th 38
theorem
for b1 being IncSpace-like IncStruct
for b2, b3, b4, b5 being Element of the Points of b1
      st {b2,b3,b4} is linear(b1)
   holds {b2,b3,b4,b5} is planar(b1);

:: INCSP_1:th 39
theorem
for b1 being IncSpace-like IncStruct
for b2, b3, b4 being Element of the Points of b1
for b5 being Element of the Lines of b1
      st b2 <> b3 & {b2,b3} on b5 & not b4 on b5
   holds {b2,b3,b4} is not linear(b1);

:: INCSP_1:th 40
theorem
for b1 being IncSpace-like IncStruct
for b2, b3, b4, b5 being Element of the Points of b1
for b6 being Element of the Planes of b1
      st {b2,b3,b4} is not linear(b1) & {b2,b3,b4} on b6 & not b5 on b6
   holds {b2,b3,b4,b5} is not planar(b1);

:: INCSP_1:th 41
theorem
for b1 being IncSpace-like IncStruct
for b2, b3 being Element of the Lines of b1
      st for b4 being Element of the Planes of b1
              st b2 on b4
           holds not b3 on b4
   holds b2 <> b3;

:: INCSP_1:th 42
theorem
for b1 being IncSpace-like IncStruct
for b2, b3, b4 being Element of the Lines of b1
      st (for b5 being Element of the Planes of b1
               st b2 on b5 & b3 on b5
            holds not b4 on b5) &
         (ex b5 being Element of the Points of b1 st
            b5 on b2 & b5 on b3 & b5 on b4)
   holds b2 <> b3;

:: INCSP_1:th 43
theorem
for b1 being IncSpace-like IncStruct
for b2, b3, b4 being Element of the Lines of b1
for b5 being Element of the Planes of b1
   st b2 on b5 & b3 on b5 & not b4 on b5 & b2 <> b3
for b6 being Element of the Planes of b1
      st b4 on b6 & b2 on b6
   holds not b3 on b6;

:: INCSP_1:th 44
theorem
for b1 being IncSpace-like IncStruct
for b2 being Element of the Points of b1
for b3 being Element of the Lines of b1 holds
   ex b4 being Element of the Planes of b1 st
      b2 on b4 & b3 on b4;

:: INCSP_1:th 45
theorem
for b1 being IncSpace-like IncStruct
for b2, b3 being Element of the Lines of b1
      st ex b4 being Element of the Points of b1 st
           b4 on b2 & b4 on b3
   holds ex b4 being Element of the Planes of b1 st
      b2 on b4 & b3 on b4;

:: INCSP_1:th 46
theorem
for b1 being IncSpace-like IncStruct
for b2, b3 being Element of the Points of b1
      st b2 <> b3
   holds ex b4 being Element of the Lines of b1 st
      for b5 being Element of the Lines of b1 holds
            {b2,b3} on b5
         iff
            b5 = b4;

:: INCSP_1:th 47
theorem
for b1 being IncSpace-like IncStruct
for b2, b3, b4 being Element of the Points of b1
      st {b2,b3,b4} is not linear(b1)
   holds ex b5 being Element of the Planes of b1 st
      for b6 being Element of the Planes of b1 holds
            {b2,b3,b4} on b6
         iff
            b5 = b6;

:: INCSP_1:th 48
theorem
for b1 being IncSpace-like IncStruct
for b2 being Element of the Points of b1
for b3 being Element of the Lines of b1
      st not b2 on b3
   holds ex b4 being Element of the Planes of b1 st
      for b5 being Element of the Planes of b1 holds
            b2 on b5 & b3 on b5
         iff
            b4 = b5;

:: INCSP_1:th 49
theorem
for b1 being IncSpace-like IncStruct
for b2, b3 being Element of the Lines of b1
      st b2 <> b3 &
         (ex b4 being Element of the Points of b1 st
            b4 on b2 & b4 on b3)
   holds ex b4 being Element of the Planes of b1 st
      for b5 being Element of the Planes of b1 holds
            b2 on b5 & b3 on b5
         iff
            b4 = b5;

:: INCSP_1:funcnot 1 => INCSP_1:func 1
definition
  let a1 be IncSpace-like IncStruct;
  let a2, a3 be Element of the Points of a1;
  assume a2 <> a3;
  func Line(A2,A3) -> Element of the Lines of a1 means
    {a2,a3} on it;
end;

:: INCSP_1:def 19
theorem
for b1 being IncSpace-like IncStruct
for b2, b3 being Element of the Points of b1
   st b2 <> b3
for b4 being Element of the Lines of b1 holds
      b4 = Line(b2,b3)
   iff
      {b2,b3} on b4;

:: INCSP_1:funcnot 2 => INCSP_1:func 2
definition
  let a1 be IncSpace-like IncStruct;
  let a2, a3, a4 be Element of the Points of a1;
  assume {a2,a3,a4} is not linear(a1);
  func Plane(A2,A3,A4) -> Element of the Planes of a1 means
    {a2,a3,a4} on it;
end;

:: INCSP_1:def 20
theorem
for b1 being IncSpace-like IncStruct
for b2, b3, b4 being Element of the Points of b1
   st {b2,b3,b4} is not linear(b1)
for b5 being Element of the Planes of b1 holds
      b5 = Plane(b2,b3,b4)
   iff
      {b2,b3,b4} on b5;

:: INCSP_1:funcnot 3 => INCSP_1:func 3
definition
  let a1 be IncSpace-like IncStruct;
  let a2 be Element of the Points of a1;
  let a3 be Element of the Lines of a1;
  assume not a2 on a3;
  func Plane(A2,A3) -> Element of the Planes of a1 means
    a2 on it & a3 on it;
end;

:: INCSP_1:def 21
theorem
for b1 being IncSpace-like IncStruct
for b2 being Element of the Points of b1
for b3 being Element of the Lines of b1
   st not b2 on b3
for b4 being Element of the Planes of b1 holds
      b4 = Plane(b2,b3)
   iff
      b2 on b4 & b3 on b4;

:: INCSP_1:funcnot 4 => INCSP_1:func 4
definition
  let a1 be IncSpace-like IncStruct;
  let a2, a3 be Element of the Lines of a1;
  assume a2 <> a3 &
     (ex b1 being Element of the Points of a1 st
        b1 on a2 & b1 on a3);
  func Plane(A2,A3) -> Element of the Planes of a1 means
    a2 on it & a3 on it;
end;

:: INCSP_1:def 22
theorem
for b1 being IncSpace-like IncStruct
for b2, b3 being Element of the Lines of b1
   st b2 <> b3 &
      (ex b4 being Element of the Points of b1 st
         b4 on b2 & b4 on b3)
for b4 being Element of the Planes of b1 holds
      b4 = Plane(b2,b3)
   iff
      b2 on b4 & b3 on b4;

:: INCSP_1:th 57
theorem
for b1 being IncSpace-like IncStruct
for b2, b3 being Element of the Points of b1
      st b2 <> b3
   holds Line(b2,b3) = Line(b3,b2);

:: INCSP_1:th 58
theorem
for b1 being IncSpace-like IncStruct
for b2, b3, b4 being Element of the Points of b1
      st {b2,b3,b4} is not linear(b1)
   holds Plane(b2,b3,b4) = Plane(b2,b4,b3);

:: INCSP_1:th 59
theorem
for b1 being IncSpace-like IncStruct
for b2, b3, b4 being Element of the Points of b1
      st {b2,b3,b4} is not linear(b1)
   holds Plane(b2,b3,b4) = Plane(b3,b2,b4);

:: INCSP_1:th 60
theorem
for b1 being IncSpace-like IncStruct
for b2, b3, b4 being Element of the Points of b1
      st {b2,b3,b4} is not linear(b1)
   holds Plane(b2,b3,b4) = Plane(b3,b4,b2);

:: INCSP_1:th 61
theorem
for b1 being IncSpace-like IncStruct
for b2, b3, b4 being Element of the Points of b1
      st {b2,b3,b4} is not linear(b1)
   holds Plane(b2,b3,b4) = Plane(b4,b2,b3);

:: INCSP_1:th 62
theorem
for b1 being IncSpace-like IncStruct
for b2, b3, b4 being Element of the Points of b1
      st {b2,b3,b4} is not linear(b1)
   holds Plane(b2,b3,b4) = Plane(b4,b3,b2);

:: INCSP_1:th 64
theorem
for b1 being IncSpace-like IncStruct
for b2, b3 being Element of the Lines of b1
      st b2 <> b3 &
         (ex b4 being Element of the Points of b1 st
            b4 on b2 & b4 on b3)
   holds Plane(b2,b3) = Plane(b3,b2);

:: INCSP_1:th 65
theorem
for b1 being IncSpace-like IncStruct
for b2, b3, b4 being Element of the Points of b1
      st b2 <> b3 & b4 on Line(b2,b3)
   holds {b2,b3,b4} is linear(b1);

:: INCSP_1:th 66
theorem
for b1 being IncSpace-like IncStruct
for b2, b3, b4 being Element of the Points of b1
      st b2 <> b3 & b2 <> b4 & {b2,b3,b4} is linear(b1)
   holds Line(b2,b3) = Line(b2,b4);

:: INCSP_1:th 67
theorem
for b1 being IncSpace-like IncStruct
for b2, b3, b4 being Element of the Points of b1
      st {b2,b3,b4} is not linear(b1)
   holds Plane(b2,b3,b4) = Plane(b4,Line(b2,b3));

:: INCSP_1:th 68
theorem
for b1 being IncSpace-like IncStruct
for b2, b3, b4, b5 being Element of the Points of b1
      st {b2,b3,b4} is not linear(b1) & b5 on Plane(b2,b3,b4)
   holds {b2,b3,b4,b5} is planar(b1);

:: INCSP_1:th 69
theorem
for b1 being IncSpace-like IncStruct
for b2, b3, b4 being Element of the Points of b1
for b5 being Element of the Lines of b1
      st not b2 on b5 & {b3,b4} on b5 & b3 <> b4
   holds Plane(b2,b5) = Plane(b3,b4,b2);

:: INCSP_1:th 70
theorem
for b1 being IncSpace-like IncStruct
for b2, b3, b4 being Element of the Points of b1
      st {b2,b3,b4} is not linear(b1)
   holds Plane(b2,b3,b4) = Plane(Line(b2,b3),Line(b2,b4));

:: INCSP_1:th 71
theorem
for b1 being IncSpace-like IncStruct
for b2 being Element of the Planes of b1 holds
   ex b3, b4, b5 being Element of the Points of b1 st
      {b3,b4,b5} on b2 & {b3,b4,b5} is not linear(b1);

:: INCSP_1:th 72
theorem
for b1 being IncSpace-like IncStruct
for b2 being Element of the Planes of b1 holds
   ex b3, b4, b5, b6 being Element of the Points of b1 st
      b3 on b2 & {b3,b4,b5,b6} is not planar(b1);

:: INCSP_1:th 73
theorem
for b1 being IncSpace-like IncStruct
for b2 being Element of the Points of b1
for b3 being Element of the Lines of b1 holds
   ex b4 being Element of the Points of b1 st
      b2 <> b4 & b4 on b3;

:: INCSP_1:th 74
theorem
for b1 being IncSpace-like IncStruct
for b2, b3 being Element of the Points of b1
for b4 being Element of the Planes of b1
      st b2 <> b3
   holds ex b5 being Element of the Points of b1 st
      b5 on b4 & {b2,b3,b5} is not linear(b1);

:: INCSP_1:th 75
theorem
for b1 being IncSpace-like IncStruct
for b2, b3, b4 being Element of the Points of b1
      st {b2,b3,b4} is not linear(b1)
   holds ex b5 being Element of the Points of b1 st
      {b2,b3,b4,b5} is not planar(b1);

:: INCSP_1:th 76
theorem
for b1 being IncSpace-like IncStruct
for b2 being Element of the Points of b1
for b3 being Element of the Planes of b1 holds
   ex b4, b5 being Element of the Points of b1 st
      {b4,b5} on b3 & {b2,b4,b5} is not linear(b1);

:: INCSP_1:th 77
theorem
for b1 being IncSpace-like IncStruct
for b2, b3 being Element of the Points of b1
      st b2 <> b3
   holds ex b4, b5 being Element of the Points of b1 st
      {b2,b3,b4,b5} is not planar(b1);

:: INCSP_1:th 78
theorem
for b1 being IncSpace-like IncStruct
for b2 being Element of the Points of b1 holds
   ex b3, b4, b5 being Element of the Points of b1 st
      {b2,b3,b4,b5} is not planar(b1);

:: INCSP_1:th 79
theorem
for b1 being IncSpace-like IncStruct
for b2 being Element of the Points of b1
for b3 being Element of the Planes of b1 holds
   ex b4 being Element of the Lines of b1 st
      not b2 on b4 & b4 on b3;

:: INCSP_1:th 80
theorem
for b1 being IncSpace-like IncStruct
for b2 being Element of the Points of b1
for b3 being Element of the Planes of b1
      st b2 on b3
   holds ex b4, b5, b6 being Element of the Lines of b1 st
      b5 <> b6 & b5 on b3 & b6 on b3 & not b4 on b3 & b2 on b4 & b2 on b5 & b2 on b6;

:: INCSP_1:th 81
theorem
for b1 being IncSpace-like IncStruct
for b2 being Element of the Points of b1 holds
   ex b3, b4, b5 being Element of the Lines of b1 st
      b2 on b3 &
       b2 on b4 &
       b2 on b5 &
       (for b6 being Element of the Planes of b1
             st b3 on b6 & b4 on b6
          holds not b5 on b6);

:: INCSP_1:th 82
theorem
for b1 being IncSpace-like IncStruct
for b2 being Element of the Points of b1
for b3 being Element of the Lines of b1 holds
   ex b4 being Element of the Planes of b1 st
      b2 on b4 & not b3 on b4;

:: INCSP_1:th 83
theorem
for b1 being IncSpace-like IncStruct
for b2 being Element of the Lines of b1
for b3 being Element of the Planes of b1 holds
   ex b4 being Element of the Points of b1 st
      b4 on b3 & not b4 on b2;

:: INCSP_1:th 84
theorem
for b1 being IncSpace-like IncStruct
for b2 being Element of the Lines of b1 holds
   ex b3 being Element of the Lines of b1 st
      for b4 being Element of the Planes of b1
            st b2 on b4
         holds not b3 on b4;

:: INCSP_1:th 85
theorem
for b1 being IncSpace-like IncStruct
for b2 being Element of the Lines of b1 holds
   ex b3, b4 being Element of the Planes of b1 st
      b3 <> b4 & b2 on b3 & b2 on b4;

:: INCSP_1:th 87
theorem
for b1 being IncSpace-like IncStruct
for b2, b3 being Element of the Points of b1
for b4 being Element of the Lines of b1
for b5 being Element of the Planes of b1
      st not b4 on b5 & {b2,b3} on b4 & {b2,b3} on b5
   holds b2 = b3;

:: INCSP_1:th 88
theorem
for b1 being IncSpace-like IncStruct
for b2, b3 being Element of the Planes of b1
      st b2 <> b3 &
         (ex b4 being Element of the Points of b1 st
            b4 on b2 & b4 on b3)
   holds ex b4 being Element of the Lines of b1 st
      for b5 being Element of the Points of b1 holds
            b5 on b2 & b5 on b3
         iff
            b5 on b4;