let i, j be Element of NAT ; :: thesis: for a being real number
for A being Matrix of REAL st [i,j] in Indices A holds
(a * A) * i,j = a * (A * i,j)
let a be real number ; :: thesis: for A being Matrix of REAL st [i,j] in Indices A holds
(a * A) * i,j = a * (A * i,j)
let A be Matrix of REAL ; :: thesis: ( [i,j] in Indices A implies (a * A) * i,j = a * (A * i,j) )
assume A1: [i,j] in Indices A ; :: thesis: (a * A) * i,j = a * (A * i,j)
reconsider aa = a as Element of F_Real by XREAL_0:def 1;
(a * A) * i,j = (MXF2MXR (aa * (MXR2MXF A))) * i,j by MATRIXR1:def 7
.= aa * ((MXR2MXF A) * i,j) by A1, MATRIX_3:def 5
.= a * (A * i,j) ;
hence (a * A) * i,j = a * (A * i,j) ; :: thesis: verum