let i, j be Nat; :: thesis: for a being Real
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; :: 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