
theorem Th4:
  for J be Function of REAL 1,REAL st J=proj(1,1) holds (for x be
  VECTOR of REAL-NS 1, y be Real st J.x=y holds ||.x.|| = |.y.|) &
(
for x,y be VECTOR of REAL-NS 1, a,b be Real st J.x=a & J.y=b holds
J
.(x+y) = a+b) & (for x be VECTOR of REAL-NS 1, y,a be Real
st J.x=y holds J.(a*x) = a*y) & (for x be VECTOR of REAL-NS 1, a be Real
 st J.x=a holds J.(-x) = -a) & for x,y be VECTOR of REAL-NS 1, a,b be
  Real st J.x=a & J.y=b holds J.(x-y) = a-b
proof
  let J be Function of REAL 1,REAL;
  assume
A1: J=proj(1,1);
  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;
    assume J.x=y;
    then I.(J.xx) = I.y;
    then x = I.y by A1,Lm1,FUNCT_1:34;
    hence ||.x.|| = |.y.| by Th3;
  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 Element of REAL by XREAL_0:def 1;
    assume that
A2: J.x=a and
A3: J.y=b;
    I.(J.yy) = I.b by A3;
    then
A4: y = I.b by A1,Lm1,FUNCT_1:34;
    I.(J.xx) = I.a by A2;
    then x = I.a by A1,Lm1,FUNCT_1:34;
    then J.(x+y) = J.(I.(a+b)) by A4,Th3;
    then J.(x+y) = J.(I.(aa+bb));
    hence J.(x+y) = a+b by A1,Lm1,FUNCT_1:35;
  end;
  hereby
    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;
    reconsider yy=y,aa=a as Element of REAL by XREAL_0:def 1;
    assume J.x=y;
    then I.(J.xx) = I.y;
    then x = I.y by A1,Lm1,FUNCT_1:34;
    then J.(a*x) = J.(I.(a*y)) by Th3;
    then J.(a*x) = aa*yy by A1,Lm1,FUNCT_1:35;
    hence J.(a*x) = a*y;
  end;
  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 Element of REAL by XREAL_0:def 1;
    assume J.x=y;
    then I.y = I.(J.xx);
    then x = I.y by A1,Lm1,FUNCT_1:34;
    then J.(-x) = J.(I.(-y)) by Th3;
    then J.(-x) = -yy by A1,Lm1,FUNCT_1:35;
    hence J.(-x) = -y;
  end;
  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 Element of REAL by XREAL_0:def 1;
  assume that
A5: J.x=a and
A6: J.y=b;
  I.(J.yy) = I.b by A6;
  then
A7: y = I.b by A1,Lm1,FUNCT_1:34;
  I.(J.xx) = I.a by A5;
  then x = I.a by A1,Lm1,FUNCT_1:34;
  then J.(x-y) =J.(I.(a - b)) by A7,Th3;
  then J.(x-y) =aa-bb by A1,Lm1,FUNCT_1:35;
  hence thesis;
end;
