reserve V for RealLinearSpace;
reserve x,y for VECTOR of V;

theorem Th1:
  Gen x,y implies (for u,u1,v,v1,w being Element of CESpace(V,x,y)
  holds ((u,u '//' v,w) & (u,v '//' w,w) & (u,v '//' u1,v1 & u,v '//' v1,u1
  implies u=v or u1=v1) & (u,v '//' u1,v1 & u,v '//' u1,w implies u,v '//' v1,w
or u,v '//' w,v1) & (u,v '//' u1,v1 implies v,u '//' v1,u1) & (u,v '//' u1,v1 &
  u,v '//' v1,w implies u,v '//' u1,w) & (u,u1 '//' v,v1 implies v,v1 '//' u,u1
  or v,v1 '//' u1,u))) & (for u,v,w being Element of CESpace(V,x,y) holds ex u1
  being Element of CESpace(V,x,y) st w<>u1 & w,u1 '//' u,v ) & for u,v,w being
Element of CESpace(V,x,y) holds ex u1 being Element of CESpace(V,x,y) st w<>u1
  & u,v '//' w,u1
proof
  assume
A1: Gen x,y;
  set S=CESpace(V,x,y);
A2: S= OrtStr (# the carrier of V,CORTE(V,x,y) #) by ANALORT:def 7;
  thus for u,u1,v,v1,w being Element of S holds u,u '//' v,w & u,v '//' w,w &
(u,v '//' u1,v1 & u,v '//' v1,u1 implies u=v or u1=v1) & (u,v '//' u1,v1 & u,v
'//' u1,w implies u,v '//' v1,w or u,v '//' w,v1) & (u,v '//' u1,v1 implies v,u
  '//' v1,u1) & (u,v '//' u1,v1 & u,v '//' v1,w implies u,v '//' u1,w) & (u,u1
  '//' v,v1 implies v,v1 '//' u,u1 or v,v1 '//' u1,u)
  proof
    let u,u1,v,v1,w be Element of S;
    reconsider u9=u,v9=v,w9=w,u19=u1,v19=v1 as Element of V by A2;
    u9,u9,v9,w9 are_COrte_wrt x,y by ANALORT:20;
    hence u,u '//' v,w by ANALORT:54;
    u9,v9,w9,w9 are_COrte_wrt x,y by ANALORT:22;
    hence u,v '//' w,w by ANALORT:54;
    u9,v9,u19,v19 are_COrte_wrt x,y & u9,v9,v19,u19 are_COrte_wrt x,y
    implies u9=v9 or u19=v19 by A1,ANALORT:30;
    hence u,v '//' u1,v1 & u,v '//' v1,u1 implies u=v or u1=v1 by ANALORT:54;
    u9,v9,u19,v19 are_COrte_wrt x,y & u9,v9,u19,w9 are_COrte_wrt x,y
implies u9,v9,v19,w9 are_COrte_wrt x,y or u9,v9,w9,v19 are_COrte_wrt x,y by A1,
ANALORT:32;
    hence
    u,v '//' u1,v1 & u,v '//' u1,w implies u,v '//' v1,w or u,v '//' w,v1
    by ANALORT:54;
    u9,v9,u19,v19 are_COrte_wrt x,y implies v9,u9,v19,u19 are_COrte_wrt x
    ,y by ANALORT:34;
    hence u,v '//' u1,v1 implies v,u '//' v1,u1 by ANALORT:54;
    u9,v9,u19,v19 are_COrte_wrt x,y & u9,v9,v19,w9 are_COrte_wrt x,y
    implies u9,v9,u19,w9 are_COrte_wrt x,y by A1,ANALORT:36;
    hence u,v '//' u1,v1 & u,v '//' v1,w implies u,v '//' u1,w by ANALORT:54;
    u9,u19,v9,v19 are_COrte_wrt x,y implies v9,v19,u9,u19 are_COrte_wrt x
    ,y or v9,v19,u19,u9 are_COrte_wrt x,y by A1,ANALORT:18;
    hence u,u1 '//' v,v1 implies v,v1 '//' u,u1 or v,v1 '//' u1,u by ANALORT:54
;
  end;
  thus for u,v,w being Element of S holds ex u1 being Element of S st w<>u1 &
  w,u1 '//' u,v
  proof
    let u,v,w be Element of S;
    reconsider u9=u,v9=v,w9=w as Element of V by A2;
    consider u19 being Element of V such that
A3: w9<>u19 and
A4: w9,u19,u9,v9 are_COrte_wrt x,y by A1,ANALORT:38;
    reconsider u1=u19 as Element of S by A2;
    w,u1 '//' u,v by A4,ANALORT:54;
    hence thesis by A3;
  end;
  let u,v,w be Element of S;
  reconsider u9=u,v9=v,w9=w as Element of V by A2;
  consider u19 being Element of V such that
A5: w9<>u19 and
A6: u9,v9,w9,u19 are_COrte_wrt x,y by A1,ANALORT:40;
  reconsider u1=u19 as Element of S by A2;
  u,v '//' w,u1 by A6,ANALORT:54;
  hence thesis by A5;
end;
