reserve x,y for set,
  i for Nat;
reserve V for non empty CLSStruct,
  u,v,v1,v2,v3 for VECTOR of V,
  A for Subset of V,
  l, l1, l2 for C_Linear_Combination of A,
  x,y,y1,y2 for set,
  a,b for Complex,
  F for FinSequence of the carrier of V,
  f for Function of the carrier of V, COMPLEX;
reserve K,L,L1,L2,L3 for C_Linear_Combination of V;
reserve e,e1,e2 for Element of C_LinComb V;

theorem Th66:
  for V being non empty CLSStruct, M being Subset of V st M is
  Affine holds M is convex
proof
  let V be non empty CLSStruct;
  let M be Subset of V;
  assume
A1: M is Affine;
  let u,v be VECTOR of V;
  let z be Complex;
  assume that
A2: ex r being Real st z=r & 0 < r & r < 1 and
A3: u in M & v in M;
  set s = 1r-z;
  consider r being Real such that
A4: z=r and
  0 < r and
  r < 1 by A2;
  s=1-r by A4;
  then (1r-s)*u+s*v in M by A1,A3;
  hence thesis;
end;
