
theorem SUBTH0:
  for V be RealNormSpace,
      V1 be SubRealNormSpace of V,
      x,y being Point of V,
      x1,y1 being Point of V1,
      a be Real
  st x = x1 & y = y1 holds
    ||.x.|| = ||.x1.||
  & x+y = x1+y1
  & a*x = a*x1
  proof
    let V be RealNormSpace,
        V1 be SubRealNormSpace of V,
        x,y be Point of V,
        x1,y1 be Point of V1,
        a be Real;
    assume
    A1: x = x1 & y = y1;
    reconsider aa = a as Element of REAL by XREAL_0:def 1;
    thus ||.x1.|| = ((the normF of V) | (the carrier of V1)).x1
                    by DUALSP01:def 16
                 .= ||.x.|| by A1,FUNCT_1:49;
    thus x1+y1 = ((the addF of V) || (the carrier of V1)) . [x1,y1]
                 by DUALSP01:def 16
              .= x+y by A1,FUNCT_1:49;
    A2: [aa,x1] in [:REAL, the carrier of V1:];
    aa*x1 = ((the Mult of V) | [:REAL, the carrier of V1:]).[a,x1]
            by DUALSP01:def 16
         .= aa*x by A1,A2,FUNCT_1:49;
    hence a*x = a*x1;
end;
