reserve X for non empty set;
reserve x for Element of X;
reserve d1,d2 for Element of X;
reserve A for BinOp of X;
reserve M for Function of [:X,X:],X;
reserve V for Ring;
reserve V1 for Subset of V;
reserve V for Algebra;
reserve V1 for Subset of V;
reserve MR for Function of [:REAL,X:],X;
reserve a for Real;

theorem Th8:
  for V be Algebra, V1 be Subalgebra of V holds
  ( for v1,w1 be Element of V1, v,w be Element of V st v1=v & w1=w
   holds v1+w1=v+w ) &
   ( for v1,  w1 be Element of V1, v,w be Element of V st v1=v & w1=w
    holds v1*w1=v*w ) & (
for v1 be Element of V1, v be Element of V,
    a be Real st v1=v holds a*v1=a*v )
  & 1_V1 = 1_V & 0.V1=0.V
proof
  let V be Algebra, V1 be Subalgebra of V;
  hereby
    let x1,y1 be Element of V1, x,y be Element of V;
    assume
A1: x1=x & y1=y;
    x1 + y1 = ((the addF of V)||the carrier of V1).[x1,y1] by Def9;
    hence x1 + y1 = x+y by A1,FUNCT_1:49;
  end;
  hereby
    let x1,y1 be Element of V1, x,y be Element of V;
    assume
A2: x1=x & y1=y;
    x1 * y1 = ((the multF of V)||the carrier of V1).[x1,y1] by Def9;
    hence x1 * y1 = x*y by A2,FUNCT_1:49;
  end;
  hereby
    let v1 be Element of V1, v be Element of V, a be Real;
    assume
A3: v1 = v;
    reconsider aa=a as Element of REAL by XREAL_0:def 1;
    aa * v1 = ((the Mult of V) | [:REAL,the carrier of V1:]).[aa,v1] by Def9;
    then aa * v1 = aa * v by A3,FUNCT_1:49;
   hence a * v1 = a * v;
  end;
  thus thesis by Def9;
end;
