
theorem Th13:
  for X be RealNormSpace for x,y,z being Element of
  Ring_of_BoundedLinearOperators(X) holds x+y = y+x & (x+y)+z = x+(y+z) & x+(0.
  Ring_of_BoundedLinearOperators(X)) = x & (ex t being Element of
Ring_of_BoundedLinearOperators(X) st x+t= 0.Ring_of_BoundedLinearOperators(X))
  & (x*y)*z = x*(y*z) & x*(1.Ring_of_BoundedLinearOperators(X)) = x & (1.
Ring_of_BoundedLinearOperators(X))*x = x & x*(y+z) = x*y + x*z & (y+z)*x = y*x
  + z*x
proof
  let X be RealNormSpace;
  let x,y,z be Element of Ring_of_BoundedLinearOperators(X);
  set RBLOP=Ring_of_BoundedLinearOperators(X);
  set BLOP=BoundedLinearOperators(X,X);
  set ADD=Add_(BoundedLinearOperators(X,X), R_VectorSpace_of_LinearOperators(X
  ,X));
  set MULT= FuncMult(X);
  set UNIT=FuncUnit(X);
  set RRL=RLSStruct (# BoundedLinearOperators(X,X), Zero_(
    BoundedLinearOperators(X,X), R_VectorSpace_of_LinearOperators(X,X)), Add_(
    BoundedLinearOperators(X,X), R_VectorSpace_of_LinearOperators(X,X)), Mult_(
    BoundedLinearOperators(X,X), R_VectorSpace_of_LinearOperators(X,X)) #);
  reconsider f=x, g=y, h=z as Element of RRL;
  thus x+y =f+g .= y+x by RLVECT_1:2;
  thus (x+y)+z =(f+g)+h .=f+(g+h) by RLVECT_1:def 3
    .= x+(y+z);
  thus x+(0.RBLOP) = f + 0.RRL .= x;
  thus ex t being Element of RBLOP st x+t=(0.RBLOP)
  proof
    consider s be Element of RRL such that
A1: f + s = 0.RRL by ALGSTR_0:def 11;
    reconsider t=s as Element of RBLOP;
    take t;
    thus thesis by A1;
  end;
  reconsider xx=x,yy=y,zz=z as Element of BLOP;
  thus (x*y)*z = MULT.(xx*yy,zz) by Def4
    .=(xx*yy)*zz by Def4
    .=xx*(yy*zz) by Th7
    .= MULT.(xx,yy*zz) by Def4
    .=x*(y*z) by Def4;
  thus x*(1.RBLOP) =xx*UNIT by Def4
    .= x by Th8;
  thus (1.RBLOP)*x =UNIT*xx by Def4
    .= x by Th8;
  thus x*(y+z) =xx*(yy+zz) by Def4
    .=xx*yy + xx*zz by Th9
    .= ADD.(xx*yy,MULT.(xx,zz)) by Def4
    .= x*y + x*z by Def4;
  thus (y+z)*x =(yy+zz)*xx by Def4
    .=yy*xx + zz*xx by Th10
    .= ADD.(yy*xx,MULT.(zz,xx)) by Def4
    .= y*x + z*x by Def4;
end;
