reserve x,y for set;
reserve X for non empty set;
reserve a,b,c,d for Element of X;
reserve S for OAffinSpace;
reserve a,b,c,d,p,q,r,x,y,z,t,u,w for Element of S;
reserve AS for non empty AffinStruct;

theorem Th39:
  AS = Lambda(S) implies (ex x,y being Element of AS st x<>y) & (
for x,y,z,t,u,w being Element of AS holds x,y // y,x & x,y // z,z & (x<>y & x,y
// z,t & x,y // u,w implies z,t // u,w) & (x,y // x,z implies y,x // y,z)) & (
  ex x,y,z being Element of AS st not x,y // x,z) & (for x,y,z being Element of
AS ex t being Element of AS st x,z // y,t & y<>t) & (for x,y,z being Element of
  AS ex t being Element of AS st x,y // z,t & x,z // y,t) & for x,y,z,t being
Element of AS st z,x // x,t & x<>z ex u being Element of AS st y,x // x,u & y,z
  // t,u
proof
  assume
A1: AS = Lambda(S);
  hence ex x,y being Element of AS st x<>y by STRUCT_0:def 10;
  thus for x,y,z,t,u,w being Element of AS holds x,y // y,x & x,y // z,z & (x
<>y & x,y // z,t & x,y // u,w implies z,t // u,w) & (x,y // x,z implies y,x //
  y,z)
  proof
    let x,y,z,t,u,w be Element of AS;
    reconsider x9=x, y9=y, z9=z, t9=t, u9=u, w9=w as Element of S by A1;
    x9,y9 '||' y9,x9 & x9,y9 '||' z9,z9 by Th19,Th20;
    hence x,y // y,x & x,y // z,z by A1,Th38;
    x9<>y9 & x9,y9 '||' z9,t9 & x9,y9 '||' u9,w9 implies z9,t9 '||' u9,w9
    by Lm2;
    hence x<>y & x,y // z,t & x,y // u,w implies z,t // u,w by A1,Th38;
    x9,y9 '||' x9,z9 implies y9,x9 '||' y9,z9 by Th21;
    hence thesis by A1,Th38;
  end;
  thus ex x,y,z being Element of AS st not x,y // x,z
  proof
    consider x9,y9,z9 being Element of S such that
A2: not x9,y9 '||' x9,z9 by Th24;
    reconsider x=x9, y=y9, z=z9 as Element of AS by A1;
    not x,y // x,z by A1,A2,Th38;
    hence thesis;
  end;
  thus for x,y,z being Element of AS ex t being Element of AS st x,z // y,t &
  y<>t
  proof
    let x,y,z be Element of AS;
    reconsider x9=x, y9=y, z9=z as Element of S by A1;
    consider t9 being Element of S such that
A3: x9,z9 '||' y9,t9 and
A4: y9<>t9 by Th25;
    reconsider t=t9 as Element of AS by A1;
    x,z // y,t by A1,A3,Th38;
    hence thesis by A4;
  end;
  thus for x,y,z being Element of AS ex t being Element of AS st x,y // z,t &
  x,z // y,t
  proof
    let x,y,z be Element of AS;
    reconsider x9=x, y9=y, z9=z as Element of S by A1;
    consider t9 being Element of S such that
A5: x9,y9 '||' z9,t9 & x9,z9 '||' y9,t9 by Th26;
    reconsider t=t9 as Element of AS by A1;
    x,y // z,t & x,z // y,t by A1,A5,Th38;
    hence thesis;
  end;
  thus for x,y,z,t being Element of AS st z,x // x,t & x<>z ex u being Element
  of AS st y,x // x,u & y,z // t,u
  proof
    let x,y,z,t be Element of AS such that
A6: z,x // x,t and
A7: x<>z;
    reconsider x9=x, y9=y, z9=z, t9=t as Element of S by A1;
    z9,x9 '||' x9,t9 by A1,A6,Th38;
    then consider u9 being Element of S such that
A8: y9,x9 '||' x9,u9 & y9,z9 '||' t9,u9 by A7,Th27;
    reconsider u=u9 as Element of AS by A1;
    y,x // x,u & y,z // t,u by A1,A8,Th38;
    hence thesis;
  end;
end;
