reserve x,y for set,
        r,s for Real,
        S for non empty addLoopStr,
        LS,LS1,LS2 for Linear_Combination of S,
        G for Abelian add-associative right_zeroed right_complementable
          non empty addLoopStr,
        LG,LG1,LG2 for Linear_Combination of G,
        g,h for Element of G,
        RLS for non empty RLSStruct,
        R for vector-distributive scalar-distributive scalar-associative
        scalar-unitalnon empty RLSStruct,
        AR for Subset of R,
        LR,LR1,LR2 for Linear_Combination of R,
        V for RealLinearSpace,
        v,v1,v2,w,p for VECTOR of V,
        A,B for Subset of V,
        F1,F2 for Subset-Family of V,
        L,L1,L2 for Linear_Combination of V;
reserve I for affinely-independent Subset of V;

theorem Th75:
  for B st x in Affin I & for y st y in B holds (x|--I).y = 0
    holds x in Affin(I\B) & x |-- I = x |-- (I\B)
 proof
  let B such that
   A1: x in Affin I and
   A2: for y st y in B holds(x|--I).y=0;
  A3: Affin I={Sum L where L is Linear_Combination of I:sum L=1} by Th59;
  A4: I\B is affinely-independent by Th43,XBOOLE_1:36;
  consider L be Linear_Combination of I such that
   A5: x=Sum L & sum L=1 by A1,A3;
  A6: x|--I=L by A1,A5,Def7;
  Carrier L c=I\B
  proof
   A7: Carrier L c=I by RLVECT_2:def 6;
   let y be object such that
    A8: y in Carrier L;
   assume not y in I\B;
   then y in B by A7,A8,XBOOLE_0:def 5;
   then L.y=0 by A2,A6;
   hence thesis by A8,RLVECT_2:19;
  end;
  then A9: L is Linear_Combination of I\B by RLVECT_2:def 6;
  then x in {Sum K where K is Linear_Combination of I\B:sum K=1} by A5;
  then x in Affin(I\B) by Th59;
  hence thesis by A4,A5,A6,A9,Def7;
 end;
