reserve a,b,c,d,e,f for Real,
        k,m for Nat,
        D for non empty set,
        V for non trivial RealLinearSpace,
        u,v,w for Element of V,
        p,q,r for Element of ProjectiveSpace(V);
reserve o,p,q,r,s,t for Point of TOP-REAL 3,
        M for Matrix of 3,F_Real;

theorem Th33:
  for M being Matrix of k,F_Real holds
  Mx2Tran M is linear-transformation of
  RLSp2RVSp(TOP-REAL k),RLSp2RVSp(TOP-REAL k)
  proof
    let M be Matrix of k,F_Real;
    reconsider M2 = Mx2Tran M as Function of RLSp2RVSp(TOP-REAL k),
      RLSp2RVSp(TOP-REAL k) by Th32;
A1: RLSp2RVSp(TOP-REAL k) = ModuleStr (# the carrier of TOP-REAL k,
      the addF of TOP-REAL k, the ZeroF of TOP-REAL k,
    MultF_Real*(TOP-REAL k) #) by DUALSP01:def 2;
    for x,y be Element of RLSp2RVSp(TOP-REAL k) holds M2.(x + y) = M2.x + M2.y
    proof
      let x,y be Element of RLSp2RVSp(TOP-REAL k);
      reconsider xr = x,yr = y as Element of TOP-REAL k by A1;
A2:   x + y = xr + yr by A1;
      M2.(x + y) = (Mx2Tran M).xr + (Mx2Tran M).yr by A2,MATRTOP1:22
                .= M2.x + M2.y by A1;
      hence thesis;
    end;
    then
A3: M2 is additive;
    for a being Scalar of F_Real, x being Vector of RLSp2RVSp(TOP-REAL k)
    holds M2.(a * x) = a * (M2 . x)
    proof
      let a be Scalar of F_Real,x be Vector of RLSp2RVSp(TOP-REAL k);
      reconsider ra = a as Real;
      reconsider rx = x as Element of TOP-REAL k by A1;
      reconsider X = RLSp2RVSp(TOP-REAL 3) as ModuleStr over F_Real;
A4:   a * x = a * rx
      proof
        MultF_Real*(TOP-REAL k) = the Mult of TOP-REAL k by DUALSP01:def 1;
        hence thesis by A1,RLVECT_1:def 1;
      end;
      a * ((Mx2Tran M).x) = a * (M2.x)
      proof
        a * (M2.x) = (the Mult of TOP-REAL k).(a,(Mx2Tran M).x)
          by A1,DUALSP01:def 1
                  .= a * ((Mx2Tran M).rx) by RLVECT_1:def 1;
        hence thesis;
      end;
      hence thesis by A4,MATRTOP1:23;
    end;
    hence thesis by A3,MOD_2:def 2;
  end;
