
theorem Th3:
  for I be Function of REAL,REAL 1 st I=proj(1,1) qua Function"
  holds (for x be VECTOR of REAL-NS 1, y be Real st
x=I.y holds ||.x.|| = |.y.|) &
(for x,y be VECTOR of REAL-NS 1, a,b be Real st x=I.a
  & y=I.b holds x+y = I.(a+b)) & (for x be VECTOR of REAL-NS 1,
y,a be Real st x=I.y holds a*x = I.(a*y)) &
(for x be VECTOR of REAL-NS 1,
a be Real st x=I.a holds -x = I.(-a)) & for x,y be
VECTOR of REAL-NS
  1, a,b be Real st x=I.a & y=I.b holds x-y =I.(a-b)
proof
  let I be Function of REAL,REAL 1;
  assume
A1: I=proj(1,1) qua Function";
  hereby
    let x be VECTOR of REAL-NS 1, y be Real;
    reconsider xx=x as Element of REAL 1 by REAL_NS1:def 4;
    reconsider yy=y as Real;
    reconsider yy2=yy^2 as Real;
    assume x = I.y;
    then xx = <*y*> by A1,Lm1;
    then sqrt Sum sqr xx = sqrt Sum <*yy2*> by RVSUM_1:55;
    then
A2: sqrt Sum sqr xx = sqrt (y^2) by RVSUM_1:73;
    ||.x.|| = |.xx.| by REAL_NS1:1;
    hence ||.x.|| = |.y.| by A2,COMPLEX1:72;
  end;
A3: now
    let x,y be VECTOR of REAL-NS 1, a,b be Real;
    reconsider xx=x, yy=y as Element of REAL 1 by REAL_NS1:def 4;
    reconsider aa=a,bb=b as Real;
    assume that
A4: x=I.a and
A5: y=I.b;
A6: <*b*> = yy by A1,A5,Lm1;
    <*a*> = xx by A1,A4,Lm1;
    then x-y = <*aa*> - <*bb*> by A6,REAL_NS1:5;
    then x-y = <*a-b*> by RVSUM_1:29;
    hence x-y =I.(a-b) by A1,Lm1;
  end;
  hereby
    let x,y be VECTOR of REAL-NS 1, a,b be Real;
    reconsider xx=x, yy = y as Element of REAL 1 by REAL_NS1:def 4;
    reconsider aa=a,bb=b as Real;
    assume that
A7: x=I.a and
A8: y=I.b;
A9: <*b*> = yy by A1,A8,Lm1;
    <*a*> = xx by A1,A7,Lm1;
    then x+y = <*aa*> + <*bb*> by A9,REAL_NS1:2;
    then x+y =<*a+b*> by RVSUM_1:13;
    hence x+y =I.(a + b) by A1,Lm1;
  end;
A10: now
    let x be VECTOR of REAL-NS 1, y,a be Real;
    reconsider xx=x as Element of REAL 1 by REAL_NS1:def 4;
    assume x=I.y;
    then
A11: xx=<*y*> by A1,Lm1;
    a*x = a*xx by REAL_NS1:3;
    then a*x =<* a*y *> by A11,RVSUM_1:47;
    hence a*x =I.(a*y) by A1,Lm1;
  end;
  now
    let x be VECTOR of REAL-NS 1, a be Real;
    assume x=I.a;
    then (-1)*x = I.((-1)*a) by A10;
    hence -x = I.(-a) by RLVECT_1:16;
  end;
  hence thesis by A10,A3;
end;
