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 Th62:
  L is convex iff sum L = 1 & for v holds 0 <= L.v
 proof
  hereby assume L is convex;
   then consider F be FinSequence of the carrier of V such that
    A1: F is one-to-one and
    A2: rng F=Carrier L and
    A3: ex f be FinSequence of REAL st len f=len F & Sum(f)=1 & for n be Nat st
n in dom f holds f.n=L.(F.n) & f.n>=0;
   consider f be FinSequence of REAL such that
    A4: len f=len F and
    A5: Sum(f)=1 and
    A6: for n be Nat st n in dom f holds f.n=L.(F.n) & f.n>=0 by A3;
   A7: len(L*F)=len F by FINSEQ_2:33;
   now let k be Nat;
    assume A8: 1<=k & k<=len F;
    then k in dom f by A4,FINSEQ_3:25;
    then A9: f.k=L.(F.k) by A6;
    k in dom(L*F) by A7,A8,FINSEQ_3:25;
    hence (L*F).k=f.k by A9,FUNCT_1:12;
   end;
   then L*F=f by A4,A7;
   hence sum L=1 by A1,A2,A5,Def3;
   let v be Element of V;
   per cases;
   suppose v in Carrier L;
    then consider n be object such that
     A10: n in dom F and
     A11: F.n=v by A2,FUNCT_1:def 3;
    A12: dom f=dom F by A4,FINSEQ_3:29;
    then f.n=L.(F.n) by A6,A10;
    hence L.v>=0 by A6,A10,A11,A12;
   end;
   suppose not v in Carrier L;
    hence L.v>=0;
   end;
  end;
  assume sum L=1;
  then consider F be FinSequence of V such that
   A13: F is one-to-one & rng F=Carrier L and
   A14: 1=Sum(L*F) by Def3;
  assume A15: for v be Element of V holds L.v>=0;
  now take F;
   thus F is one-to-one & rng F=Carrier L by A13;
   take f=L*F;
   thus len f=len F & Sum f=1 by A14,FINSEQ_2:33;
   then A16: dom F=dom f by FINSEQ_3:29;
   let n be Nat;
   assume A17: n in dom f;
   then A18: f.n=L.(F.n) by FUNCT_1:12;
   F.n=F/.n by A16,A17,PARTFUN1:def 6;
   hence f.n=L.(F.n) & f.n>=0 by A15,A18;
  end;
  hence thesis;
 end;
